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83-465: The Method may refer to: Science [ edit ] The Method of Mechanical Theorems , a work of Archimedes Discourse on the Method , a work of Descartes Arts and entertainment [ edit ] The Method (TV series) , a 2015 Russian television drama series The Method (album) , a 1997 album by Killing Time The Method (film) ,

166-629: A closed and bounded interval [ a , b ] and can be generalized to other notions of integral (Lebesgue and Daniell). In this section, f is a real-valued Riemann-integrable function . The integral over an interval [ a , b ] is defined if a < b . This means that the upper and lower sums of the function f are evaluated on a partition a = x 0 ≤ x 1 ≤ . . . ≤ x n = b whose values x i are increasing. Geometrically, this signifies that integration takes place "left to right", evaluating f within intervals [ x i  , x i  +1 ] where an interval with

249-522: A sphere , area of an ellipse , the area under a parabola , the volume of a segment of a paraboloid of revolution, the volume of a segment of a hyperboloid of revolution, and the area of a spiral . A similar method was independently developed in China around the 3rd century AD by Liu Hui , who used it to find the area of the circle. This method was later used in the 5th century by Chinese father-and-son mathematicians Zu Chongzhi and Zu Geng to find

332-461: A , b ] is its width, b − a , so that the Lebesgue integral agrees with the (proper) Riemann integral when both exist. In more complicated cases, the sets being measured can be highly fragmented, with no continuity and no resemblance to intervals. Using the "partitioning the range of f " philosophy, the integral of a non-negative function f  : R → R should be the sum over t of

415-458: A 2005 film by Marcelo Piñeyro The Method (novel) , a 2009 novel by Juli Zeh Andwella or The Method, a band of UK/Irish origin "The Method", a song by We Are Scientists from Safety, Fun, and Learning (In That Order) Method acting Stanislavski's system See also [ edit ] Method (disambiguation) Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with

498-411: A bounded interval, subsequently more general functions were considered—particularly in the context of Fourier analysis —to which Riemann's definition does not apply, and Lebesgue formulated a different definition of integral , founded in measure theory (a subfield of real analysis ). Other definitions of integral, extending Riemann's and Lebesgue's approaches, were proposed. These approaches based on

581-453: A certain class of "simple" functions, may be used to give an alternative definition of the integral. This is the approach of Daniell for the case of real-valued functions on a set X , generalized by Nicolas Bourbaki to functions with values in a locally compact topological vector space. See Hildebrandt 1953 for an axiomatic characterization of the integral. A number of general inequalities hold for Riemann-integrable functions defined on

664-453: A connection between integration and differentiation . Barrow provided the first proof of the fundamental theorem of calculus . Wallis generalized Cavalieri's method, computing integrals of x to a general power, including negative powers and fractional powers. The major advance in integration came in the 17th century with the independent discovery of the fundamental theorem of calculus by Leibniz and Newton . The theorem demonstrates

747-565: A connection between integration and differentiation. This connection, combined with the comparative ease of differentiation, can be exploited to calculate integrals. In particular, the fundamental theorem of calculus allows one to solve a much broader class of problems. Equal in importance is the comprehensive mathematical framework that both Leibniz and Newton developed. Given the name infinitesimal calculus, it allowed for precise analysis of functions with continuous domains. This framework eventually became modern calculus , whose notation for integrals

830-434: A distance x from the fulcrum on the other side. This means that the cone and the sphere together, if all their material were moved to x = 1, would balance a cylinder of base radius 1 and length 2 on the other side. As x ranges from 0 to 2, the cylinder will have a center of gravity a distance 1 from the fulcrum, so all the weight of the cylinder can be considered to be at position 1. The condition of balance ensures that

913-519: A function f over the interval [ a , b ] is equal to S if: When the chosen tags are the maximum (respectively, minimum) value of the function in each interval, the Riemann sum becomes an upper (respectively, lower) Darboux sum , suggesting the close connection between the Riemann integral and the Darboux integral . It is often of interest, both in theory and applications, to be able to pass to

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996-458: A function f with respect to such a tagged partition is defined as thus each term of the sum is the area of a rectangle with height equal to the function value at the chosen point of the given sub-interval, and width the same as the width of sub-interval, Δ i = x i − x i −1 . The mesh of such a tagged partition is the width of the largest sub-interval formed by the partition, max i =1... n Δ i . The Riemann integral of

1079-402: A higher index lies to the right of one with a lower index. The values a and b , the end-points of the interval , are called the limits of integration of f . Integrals can also be defined if a > b : With a = b , this implies: The first convention is necessary in consideration of taking integrals over subintervals of [ a , b ] ; the second says that an integral taken over

1162-414: A letter to Paul Montel : I have to pay a certain sum, which I have collected in my pocket. I take the bills and coins out of my pocket and give them to the creditor in the order I find them until I have reached the total sum. This is the Riemann integral. But I can proceed differently. After I have taken all the money out of my pocket I order the bills and coins according to identical values and then I pay

1245-416: A lever, is proportional to the area: π ρ S ( x ) 2 = 2 π x − π x 2 . {\displaystyle \pi \rho _{S}(x)^{2}=2\pi x-\pi x^{2}.} Archimedes then considered rotating the triangular region between y  = 0 and y  =  x and x = 2 on the x - y plane around

1328-416: A mass equal to its height x {\displaystyle x} , and is at a distance x {\displaystyle x} from the fulcrum; so it would balance the corresponding slice of the parabola, of height x 2 {\displaystyle x^{2}} , if the latter were moved to x = − 1 {\displaystyle x=-1} , at a distance of 1 on

1411-540: A rigorous definition of integrals, which is based on a limiting procedure that approximates the area of a curvilinear region by breaking the region into infinitesimally thin vertical slabs. In the early 20th century, Henri Lebesgue generalized Riemann's formulation by introducing what is now referred to as the Lebesgue integral ; it is more general than Riemann's in the sense that a wider class of functions are Lebesgue-integrable. Integrals may be generalized depending on

1494-458: A suitable class of functions (the measurable functions ) this defines the Lebesgue integral. A general measurable function f is Lebesgue-integrable if the sum of the absolute values of the areas of the regions between the graph of f and the x -axis is finite: In that case, the integral is, as in the Riemannian case, the difference between the area above the x -axis and the area below

1577-443: Is a lever, with a fulcrum at x = 0 {\displaystyle x=0} . The law of the lever states that two objects on opposite sides of the fulcrum will balance if each has the same torque , where an object's torque equals its weight times its distance to the fulcrum. For each value of x {\displaystyle x} , the slice of the triangle at position x {\displaystyle x} has

1660-451: Is above the point x = 2 / 3 {\displaystyle x=2/3} , so that the total effect of the triangle on the lever is as if the total mass of the triangle were pushing down on (or hanging from) this point. The total torque exerted by the triangle is its area, 1/2, times the distance 2/3 of its center of mass from the fulcrum at x = 0 {\displaystyle x=0} . This torque of 1/3 balances

1743-422: Is at the point I on the "lever" where DI  : DB  = 1:3. Therefore, it suffices to show that if the whole weight of the interior of the triangle rests at I , and the whole weight of the section of the parabola at J , the lever is in equilibrium. Consider an infinitely small cross-section of the triangle given by the segment HE , where the point H lies on BC , the point E lies on AB , and HE

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1826-418: Is defined in terms of Riemann sums of functions with respect to tagged partitions of an interval. A tagged partition of a closed interval [ a , b ] on the real line is a finite sequence This partitions the interval [ a , b ] into n sub-intervals [ x i −1 , x i ] indexed by i , each of which is "tagged" with a specific point t i ∈ [ x i −1 , x i ] . A Riemann sum of

1909-507: Is drawn directly from the work of Leibniz. While Newton and Leibniz provided a systematic approach to integration, their work lacked a degree of rigour . Bishop Berkeley memorably attacked the vanishing increments used by Newton, calling them " ghosts of departed quantities ". Calculus acquired a firmer footing with the development of limits . Integration was first rigorously formalized, using limits, by Riemann . Although all bounded piecewise continuous functions are Riemann-integrable on

1992-501: Is not uncommon to leave out dx when only the simple Riemann integral is being used, or the exact type of integral is immaterial. For instance, one might write ∫ a b ( c 1 f + c 2 g ) = c 1 ∫ a b f + c 2 ∫ a b g {\textstyle \int _{a}^{b}(c_{1}f+c_{2}g)=c_{1}\int _{a}^{b}f+c_{2}\int _{a}^{b}g} to express

2075-449: Is of great importance to have a definition of the integral that allows a wider class of functions to be integrated. Such an integral is the Lebesgue integral, that exploits the following fact to enlarge the class of integrable functions: if the values of a function are rearranged over the domain, the integral of a function should remain the same. Thus Henri Lebesgue introduced the integral bearing his name, explaining this integral thus in

2158-465: Is one of the major surviving works of the ancient Greek polymath Archimedes . The Method takes the form of a letter from Archimedes to Eratosthenes , the chief librarian at the Library of Alexandria , and contains the first attested explicit use of indivisibles (indivisibles are geometric versions of infinitesimals ). The work was originally thought to be lost, but in 1906 was rediscovered in

2241-437: Is parallel to the axis of symmetry of the parabola. Call the intersection of HE and the parabola F and the intersection of HE and the lever G . If the weight of all such segments HE rest at the points G where they intersect the lever, then they exert the same torque on the lever as does the whole weight of the triangle resting at  I . Thus, we wish to show that if the weight of the cross-section HE rests at G and

2324-449: Is placed with its center at x  = 1, the vertical cross sectional radius ρ S {\displaystyle \rho _{S}} at any x between 0 and 2 is given by the following formula: ρ S ( x ) = x ( 2 − x ) . {\displaystyle \rho _{S}(x)={\sqrt {x(2-x)}}.} The mass of this cross section, for purposes of balancing on

2407-421: Is the method of exhaustion of the ancient Greek astronomer Eudoxus and philosopher Democritus ( ca. 370 BC), which sought to find areas and volumes by breaking them up into an infinite number of divisions for which the area or volume was known. This method was further developed and employed by Archimedes in the 3rd century BC and used to calculate the area of a circle , the surface area and volume of

2490-574: Is the interior of a right triangle of side length 1 − x 2 {\displaystyle {\sqrt {1-x^{2}}}} whose area is 1 2 ( 1 − x 2 ) {\displaystyle {1 \over 2}(1-x^{2})} , so that the total volume is: ∫ − 1 1 1 2 ( 1 − x 2 ) d x {\displaystyle \displaystyle \int _{-1}^{1}{1 \over 2}(1-x^{2})\,dx} which can be easily rectified using

2573-438: Is the region in the same plane between the x {\displaystyle x} -axis and the line y = x {\displaystyle y=x} , also as x {\displaystyle x} varies from 0 to 1. Slice the parabola and triangle into vertical slices, one for each value of x {\displaystyle x} . Imagine that the x {\displaystyle x} -axis

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2656-430: The differential of the variable x , indicates that the variable of integration is x . The function f ( x ) is called the integrand, the points a and b are called the limits (or bounds) of integration, and the integral is said to be over the interval [ a , b ] , called the interval of integration. A function is said to be integrable if its integral over its domain is finite. If limits are specified,

2739-413: The real line . Conventionally, areas above the horizontal axis of the plane are positive while areas below are negative. Integrals also refer to the concept of an antiderivative , a function whose derivative is the given function; in this case, they are also called indefinite integrals . The fundamental theorem of calculus relates definite integration to differentiation and provides a method to compute

2822-483: The x -axis, to form a cone. The cross section of this cone is a circle of radius ρ C {\displaystyle \rho _{C}} ρ C ( x ) = x {\displaystyle \rho _{C}(x)=x} and the area of this cross section is π ρ C 2 = π x 2 . {\displaystyle \pi \rho _{C}^{2}=\pi x^{2}.} So if slices of

2905-430: The x -axis: where Although the Riemann and Lebesgue integrals are the most widely used definitions of the integral, a number of others exist, including: The collection of Riemann-integrable functions on a closed interval [ a , b ] forms a vector space under the operations of pointwise addition and multiplication by a scalar, and the operation of integration is a linear functional on this vector space. Thus,

2988-570: The Archimedean method mechanically balances the parabola (the curved region being integrated above) with a certain triangle that is made of the same material. The parabola is the region in the ( x , y ) {\displaystyle (x,y)} plane between the x {\displaystyle x} -axis and the curve y = x 2 {\displaystyle y=x^{2}} as x {\displaystyle x} varies from 0 to 1. The triangle

3071-492: The Equilibrium of Planes . So the center of mass of a triangle must be at the intersection point of the medians. For the triangle in question, one median is the line y = x / 2 {\displaystyle y=x/2} , while a second median is the line y = 1 − x {\displaystyle y=1-x} . Solving these equations, we see that the intersection of these two medians

3154-477: The accompanying figure of the balanced sphere, cone, and cylinder be engraved upon his tombstone. To find the surface area of the sphere, Archimedes argued that just as the area of the circle could be thought of as infinitely many infinitesimal right triangles going around the circumference (see Measurement of the Circle ), the volume of the sphere could be thought of as divided into many cones with height equal to

3237-442: The areas between a thin horizontal strip between y = t and y = t + dt . This area is just μ { x  : f ( x ) > t }  dt . Let f ( t ) = μ { x  : f ( x ) > t } . The Lebesgue integral of f is then defined by where the integral on the right is an ordinary improper Riemann integral ( f is a strictly decreasing positive function, and therefore has a well-defined improper Riemann integral). For

3320-448: The base area up, and then each cone makes a contribution according to its base area, just the same as in the sphere. Let the surface of the sphere be  S . The volume of the cone with base area S and height r is S r / 3 {\displaystyle Sr/3} , which must equal the volume of the sphere: 4 π r 3 / 3 {\displaystyle 4\pi r^{3}/3} . Therefore,

3403-481: The box notation was difficult for printers to reproduce, so these notations were not widely adopted. The term was first printed in Latin by Jacob Bernoulli in 1690: "Ergo et horum Integralia aequantur". In general, the integral of a real-valued function f ( x ) with respect to a real variable x on an interval [ a , b ] is written as The integral sign ∫ represents integration. The symbol dx , called

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3486-471: The celebrated Archimedes Palimpsest . The palimpsest includes Archimedes' account of the "mechanical method", so called because it relies on the center of weights of figures ( centroid ) and the law of the lever , which were demonstrated by Archimedes in On the Equilibrium of Planes . Archimedes did not admit the method of indivisibles as part of rigorous mathematics, and therefore did not publish his method in

3569-406: The circular prism, which is the region obeying: x 2 + y 2 < 1 , 0 < z < y . {\displaystyle x^{2}+y^{2}<1,\quad 0<z<y.} Both problems have a slicing which produces an easy integral for the mechanical method. For the circular prism, cut up the x -axis into slices. The region in the y - z plane at any x

3652-470: The collection of integrable functions is closed under taking linear combinations , and the integral of a linear combination is the linear combination of the integrals: Similarly, the set of real -valued Lebesgue-integrable functions on a given measure space E with measure μ is closed under taking linear combinations and hence form a vector space, and the Lebesgue integral is a linear functional on this vector space, so that: More generally, consider

3735-406: The cone and the sphere both are to be weighed together, the combined cross-sectional area is: M ( x ) = 2 π x . {\displaystyle M(x)=2\pi x.} If the two slices are placed together at distance 1 from the fulcrum, their total weight would be exactly balanced by a circle of area 2 π {\displaystyle 2\pi } at

3818-453: The cone from the volume of the cylinder gives the volume of the sphere: V S = 4 π − 8 3 π = 4 3 π . {\displaystyle V_{S}=4\pi -{8 \over 3}\pi ={4 \over 3}\pi .} The dependence of the volume of the sphere on the radius is obvious from scaling, although that also was not trivial to make rigorous back then. The method then gives

3901-482: The definite integral of a function when its antiderivative is known; differentiation and integration are inverse operations. Although methods of calculating areas and volumes dated from ancient Greek mathematics , the principles of integration were formulated independently by Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century, who thought of the area under a curve as an infinite sum of rectangles of infinitesimal width. Bernhard Riemann later gave

3984-546: The definite integral, with limits above and below the integral sign, was first used by Joseph Fourier in Mémoires of the French Academy around 1819–1820, reprinted in his book of 1822. Isaac Newton used a small vertical bar above a variable to indicate integration, or placed the variable inside a box. The vertical bar was easily confused with . x or x ′ , which are used to indicate differentiation, and

4067-406: The familiar formula for the volume of a sphere . By scaling the dimensions linearly Archimedes easily extended the volume result to spheroids . Archimedes argument is nearly identical to the argument above, but his cylinder had a bigger radius, so that the cone and the cylinder hung at a greater distance from the fulcrum. He considered this argument to be his greatest achievement, requesting that

4150-404: The formal treatises that contain the results. In these treatises, he proves the same theorems by exhaustion , finding rigorous upper and lower bounds which both converge to the answer required. Nevertheless, the mechanical method was what he used to discover the relations for which he later gave rigorous proofs. Archimedes' idea is to use the law of the lever to determine the areas of figures from

4233-461: The foundations of modern calculus, with Cavalieri computing the integrals of x up to degree n = 9 in Cavalieri's quadrature formula . The case n = −1 required the invention of a function , the hyperbolic logarithm , achieved by quadrature of the hyperbola in 1647. Further steps were made in the early 17th century by Barrow and Torricelli , who provided the first hints of

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4316-404: The integral involved is exactly the same as the one for area of the parabola. The volume of the cone is 1/3 its base area times the height. The base of the cone is a circle of radius 2, with area 4 π {\displaystyle 4\pi } , while the height is 2, so the area is 8 π / 3 {\displaystyle 8\pi /3} . Subtracting the volume of

4399-534: The integral is called a definite integral. When the limits are omitted, as in the integral is called an indefinite integral, which represents a class of functions (the antiderivative ) whose derivative is the integrand. The fundamental theorem of calculus relates the evaluation of definite integrals to indefinite integrals. There are several extensions of the notation for integrals to encompass integration on unbounded domains and/or in multiple dimensions (see later sections of this article). In advanced settings, it

4482-409: The integral of x 3 {\displaystyle x^{3}} , which he used to find the center of mass of a hemisphere, and in other work, the center of mass of a parabola. Consider the parabola in the figure to the right. Pick two points on the parabola and call them A and B . Suppose the line segment AC is parallel to the axis of symmetry of the parabola. Further suppose that

4565-424: The known center of mass of other figures. The simplest example in modern language is the area of the parabola. A modern approach would be to find this area by calculating the integral ∫ 0 1 x 2 d x = 1 3 , {\displaystyle \int _{0}^{1}x^{2}\,dx={\frac {1}{3}},} which is an elementary result in integral calculus . Instead,

4648-429: The limit under the integral. For instance, a sequence of functions can frequently be constructed that approximate, in a suitable sense, the solution to a problem. Then the integral of the solution function should be the limit of the integrals of the approximations. However, many functions that can be obtained as limits are not Riemann-integrable, and so such limit theorems do not hold with the Riemann integral. Therefore, it

4731-410: The line segment BC lies on a line that is tangent to the parabola at B . The first proposition states: Let D be the midpoint of AC . Construct a line segment JB through D , where the distance from J to D is equal to the distance from B to D . We will think of the segment JB as a "lever" with D as its fulcrum. As Archimedes had previously shown, the center of mass of the triangle

4814-434: The linearity holds for the subspace of functions whose integral is an element of V (i.e. "finite"). The most important special cases arise when K is R , C , or a finite extension of the field Q p of p-adic numbers , and V is a finite-dimensional vector space over K , and when K = C and V is a complex Hilbert space . Linearity, together with some natural continuity properties and normalization for

4897-518: The linearity of the integral, a property shared by the Riemann integral and all generalizations thereof. Integrals appear in many practical situations. For instance, from the length, width and depth of a swimming pool which is rectangular with a flat bottom, one can determine the volume of water it can contain, the area of its surface, and the length of its edge. But if it is oval with a rounded bottom, integrals are required to find exact and rigorous values for these quantities. In each case, one may divide

4980-399: The mechanical method. Adding to each triangular section a section of a triangular pyramid with area x 2 / 2 {\displaystyle x^{2}/2} balances a prism whose cross section is constant. For the intersection of two cylinders, the slicing is lost in the manuscript, but it can be reconstructed in an obvious way in parallel to the rest of the document: if

5063-427: The median, considered as a fulcrum. The reason is that if the triangle is divided into infinitesimal line segments parallel to E {\displaystyle E} , each segment has equal length on opposite sides of the median, so balance follows by symmetry. This argument can be easily made rigorous by exhaustion by using little rectangles instead of infinitesimal lines, and this is what Archimedes does in On

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5146-685: The number of pieces increases to infinity, it will reach a limit which is the exact value of the area sought (in this case, 2/3 ). One writes which means 2/3 is the result of a weighted sum of function values, √ x , multiplied by the infinitesimal step widths, denoted by dx , on the interval [0, 1] . There are many ways of formally defining an integral, not all of which are equivalent. The differences exist mostly to deal with differing special cases which may not be integrable under other definitions, but are also occasionally for pedagogical reasons. The most commonly used definitions are Riemann integrals and Lebesgue integrals. The Riemann integral

5229-650: The other examples, the volume of these shapes is not rigorously computed in any of his other works. From fragments in the palimpsest, it appears that Archimedes did inscribe and circumscribe shapes to prove rigorous bounds for the volume, although the details have not been preserved. The two shapes he considers are the intersection of two cylinders at right angles (the bicylinder ), which is the region of ( x ,  y ,  z ) obeying: x 2 + y 2 < 1 , y 2 + z 2 < 1 , {\displaystyle x^{2}+y^{2}<1,\quad y^{2}+z^{2}<1,} and

5312-439: The other side of the fulcrum. Since each pair of slices balances, moving the whole parabola to x = − 1 {\displaystyle x=-1} would balance the whole triangle. This means that if the original uncut parabola is hung by a hook from the point x = − 1 {\displaystyle x=-1} (so that the whole mass of the parabola is attached to that point), it will balance

5395-422: The palimpsest by similar arguments. One theorem is that the location of a center of mass of a hemisphere is located 5/8 of the way from the pole to the center of the sphere. This problem is notable, because it is evaluating a cubic integral. Integral calculus A definite integral computes the signed area of the region in the plane that is bounded by the graph of a given function between two points in

5478-436: The parabola, which is at a distance 1 from the fulcrum. Hence, the area of the parabola must be 1/3 to give it the opposite torque. This type of method can be used to find the area of an arbitrary section of a parabola, and similar arguments can be used to find the integral of any power of x {\displaystyle x} , although higher powers become complicated without algebra. Archimedes only went as far as

5561-438: The radius and base on the surface. The cones all have the same height, so their volume is 1/3 the base area times the height. Archimedes states that the total volume of the sphere is equal to the volume of a cone whose base has the same surface area as the sphere and whose height is the radius. There are no details given for the argument, but the obvious reason is that the cone can be divided into infinitesimal cones by splitting

5644-477: The real number system are the ones most common today, but alternative approaches exist, such as a definition of integral as the standard part of an infinite Riemann sum, based on the hyperreal number system. The notation for the indefinite integral was introduced by Gottfried Wilhelm Leibniz in 1675. He adapted the integral symbol , ∫ , from the letter ſ ( long s ), standing for summa (written as ſumma ; Latin for "sum" or "total"). The modern notation for

5727-417: The results to carry out what would now be called an integration of this function, where the formulae for the sums of integral squares and fourth powers allowed him to calculate the volume of a paraboloid . The next significant advances in integral calculus did not begin to appear until the 17th century. At this time, the work of Cavalieri with his method of indivisibles , and work by Fermat , began to lay

5810-409: The right end height of each piece (thus √ 0 , √ 1/5 , √ 2/5 , ..., √ 1 ) and sum their areas to get the approximation which is larger than the exact value. Alternatively, when replacing these subintervals by ones with the left end height of each piece, the approximation one gets is too low: with twelve such subintervals the approximated area is only 0.6203. However, when

5893-441: The several heaps one after the other to the creditor. This is my integral. As Folland puts it, "To compute the Riemann integral of f , one partitions the domain [ a , b ] into subintervals", while in the Lebesgue integral, "one is in effect partitioning the range of f ". The definition of the Lebesgue integral thus begins with a measure , μ. In the simplest case, the Lebesgue measure μ ( A ) of an interval A = [

5976-427: The shapes having curvilinear boundaries. This is a central point of the investigation—certain curvilinear shapes could be rectified by ruler and compass, so that there are nontrivial rational relations between the volumes defined by the intersections of geometrical solids. Archimedes emphasizes this in the beginning of the treatise, and invites the reader to try to reproduce the results by some other method. Unlike

6059-421: The sought quantity into infinitely many infinitesimal pieces, then sum the pieces to achieve an accurate approximation. As another example, to find the area of the region bounded by the graph of the function f ( x ) = x {\textstyle {\sqrt {x}}} between x = 0 and x = 1 , one can divide the interval into five pieces ( 0, 1/5, 2/5, ..., 1 ), then construct rectangles using

6142-551: The surface area of the sphere must be 4 π r 2 {\displaystyle 4\pi r^{2}} , or "four times its largest circle". Archimedes proves this rigorously in On the Sphere and Cylinder . One of the remarkable things about the Method is that Archimedes finds two shapes defined by sections of cylinders, whose volume does not involve π {\displaystyle \pi } , despite

6225-609: The title The Method . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=The_Method&oldid=1114699690 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages The Method of Mechanical Theorems The Method of Mechanical Theorems ( Greek : Περὶ μηχανικῶν θεωρημάτων πρὸς Ἐρατοσθένη ἔφοδος ), also referred to as The Method ,

6308-462: The total volume is: ∫ − 1 1 4 ( 1 − y 2 ) d y . {\displaystyle \displaystyle \int _{-1}^{1}4(1-y^{2})\,dy.} And this is the same integral as for the previous example. Jan Hogendijk argues that, besides the volume of the bicylinder, Archimedes knew its surface area , which is also rational. A series of propositions of geometry are proved in

6391-423: The triangle sitting between x = 0 {\displaystyle x=0} and x = 1 {\displaystyle x=1} . The center of mass of a triangle can be easily found by the following method, also due to Archimedes. If a median line is drawn from any one of the vertices of a triangle to the opposite edge E {\displaystyle E} , the triangle will balance on

6474-440: The type of the function as well as the domain over which the integration is performed. For example, a line integral is defined for functions of two or more variables, and the interval of integration is replaced by a curve connecting two points in space. In a surface integral , the curve is replaced by a piece of a surface in three-dimensional space . The first documented systematic technique capable of determining integrals

6557-414: The vector space of all measurable functions on a measure space ( E , μ ) , taking values in a locally compact complete topological vector space V over a locally compact topological field K , f  : E → V . Then one may define an abstract integration map assigning to each function f an element of V or the symbol ∞ , that is compatible with linear combinations. In this situation,

6640-677: The volume of a sphere. In the Middle East, Hasan Ibn al-Haytham, Latinized as Alhazen ( c.  965  – c.  1040  AD) derived a formula for the sum of fourth powers . Alhazen determined the equations to calculate the area enclosed by the curve represented by y = x k {\displaystyle y=x^{k}} (which translates to the integral ∫ x k d x {\displaystyle \int x^{k}\,dx} in contemporary notation), for any given non-negative integer value of k {\displaystyle k} . He used

6723-411: The volume of the cone plus the volume of the sphere is equal to the volume of the cylinder. The volume of the cylinder is the cross section area, 2 π {\displaystyle 2\pi } times the height, which is 2, or 4 π {\displaystyle 4\pi } . Archimedes could also find the volume of the cone using the mechanical method, since, in modern terms,

6806-416: The weight of the cross-section EF of the section of the parabola rests at J , then the lever is in equilibrium. In other words, it suffices to show that EF  : GD  =  EH  : JD . But that is a routine consequence of the equation of the parabola.  Q.E.D. Again, to illuminate the mechanical method, it is convenient to use a little bit of coordinate geometry. If a sphere of radius 1

6889-514: The x-z plane is the slice direction, the equations for the cylinder give that x 2 < 1 − y 2 {\displaystyle x^{2}<1-y^{2}} while z 2 < 1 − y 2 {\displaystyle z^{2}<1-y^{2}} , which defines a region which is a square in the x - z plane of side length 2 1 − y 2 {\displaystyle 2{\sqrt {1-y^{2}}}} , so that

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