Calculus: Analytic Geometry and Calculus, with Vectors

686 Iterated and multiple integrals Changing the variable of integration by setting u = t/(n - 1) then gives 1 z+1 n-1 t n 1 (6) ...

13.5 Integrals in polar coordinates 687 and putting x = ut gives the second equality Therefore (6) p!4! = (p + q + 1)! 101 tP(1- ...

688 Iterated and multiple integrals ferent ways of proving that, when p > 0, (13.532) lim As = 1. AP-+o, m-+o PD P A¢ It is, ...

13.5 Integrals in polar coordinates 689 Examplel Let05a<#5,r,letf(4)>_Owhen a<¢_<S,letf be Riemann integrable over t ...

690 Iterated and multiple integrals It is particularly easy to evaluate this integral for the special case in which there is a c ...

13.5 Integrals in polar coordinates 691 gives the approximation (13.58) S(P,0)P3 AP 0O to the polar moment of the subset. This l ...

692 Iterated and multiple integrals appearing among the problems of Section 8.4, and the fact that (-i)! _ /jr- we can put the r ...

13.5 Integrals in polar coordinates 693 7r and p > a1/cos 4> if 101 < it/2. It follows that if P lies outside our solid ...

694 Iterated and multiple integrals turns out to be useful because the double integral can be compared with other double integra ...

13.6 Triple integrals; rectangular coordinates 695 of agriculture are required to study analytic geometry and calculus so they c ...

696 Iterated and multiple integrals whenever the sum is a Riemann sum formed for the function f and for a partition Q of S for w ...

13.6 Triple integrals; rectangular coordinates 697 x=a1 x=a2 Y=gi(x) z=f1(x>Y) Y=g,(x) Figure 13.64 we consider an example. A ...

698 Iterated and multiple integrals We therefore use this number as an approximation to the pth moment of the subset. The sum (1 ...

13.6 Triple integrals; rectangular coordinates 699 Thus (13.653) and (13.654) Ox a:(x> dy 9,(x) )PS x z A 1,(.,v) x - xo (,y, ...

(^700) Iterated and multiple integrals and the parallel axis theorem (Theorem 13.48) hold for solidsas well as for lamina. Probl ...

13.6 Triple integrals; rectangular coordinates^701 Remark: Section 13.8 will enable us to avoid this and some other unpleasant i ...

702 Iterated and multiple integrals where S is the cube having four of its vertices at the points (0,0,0), (a,0,0), (0,a,0), (0, ...

13.7 Triple integrals; cylindrical coordinates 703 to put (13.71) in the form (13.74) fff3f(p,O,z)p do dp dz= limIf(P,o,z)P Li4 ...

704 Iterated and multiple integrals the square of the distance from the subset to the x axis, to obtain the expression (13.763) ...

13.7 Triple integrals; cylindrical coordinates 705 (^4) Assuming that the solid cone shown in the upper part of Figure 13.791 ha ...