. Differential and integral calculus, an introductory course for colleges and engineering schools. y) A/2, R where R is written under the ]v to indicate that the summation isto extend over the entire region R. Now let the number of poly-. §214 MULTIPLE INTEGRALS 323 gons be indefinitely increased, but in such a way that the greatestdiameter and consequently the area of each small polygon shallhave the limit 0. Then the fundamental theorem of the IntegralCalculus of Functions of Two Arguments is j£jf(pc9 y)AR has a limit, and this limit is independent of the R method of subdividing R into small
. Differential and integral calculus, an introductory course for colleges and engineering schools. y) A/2, R where R is written under the ]v to indicate that the summation isto extend over the entire region R. Now let the number of poly-. §214 MULTIPLE INTEGRALS 323 gons be indefinitely increased, but in such a way that the greatestdiameter and consequently the area of each small polygon shallhave the limit 0. Then the fundamental theorem of the IntegralCalculus of Functions of Two Arguments is j£jf(pc9 y)AR has a limit, and this limit is independent of the R method of subdividing R into small polygons. This theorem is an analytic one, but we shall give a geometricproof of it. On R as a base we construct a truncated right cylinder like thatof the preceding article, whose upper base lies in the surface z =f(x, y). On each of the small poly-gons AR we construct small trun-cated right cylinders.* Plainly, thelarge cylinder on base R is the sumof the small cylinders. Consider oneof these small truncated cylindersABC DM, and the corresponding pro-duct f(xk,yk)ARk. This product isthe volume of a cylinder not trun-cated, ABCDM, whose base is ARkand altitude f(xk, yk), and which co-incides with the truncated cylinderexcept i
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectcalculu, bookyear1912
