Calculus: Analytic Geometry and Calculus, with Vectors

666 reduces to (4) Iterated and multiple integrals f b dx f (z) d Y) f F(x,y) dy = f dy pz f F(x,y) dx. a !3(z) c 0i(Y) Finally, ...

13.3 Double integrals 667 Since 0! = 1, the formula (3) M-) r x (x = a (n - 1)! f(t)dt is certainly correct when n = 1. Assuming ...

668 Iterated and multiple integrals polar, for example) to determine a point P of S, we do not at present allow any one brand of ...

13.3 Double integrals 669 there corresponds a S > 0 such that (13.34) nn I-k 1 f (Pk) Ask I< whenever the sum is a Riemann ...

(^670) Iterated and multiple integrals Proof of this theorem lies far beyond the scope of this course. Persons who continue stud ...

13.3 Double integrals 671 Then show that f f f (,,y) dS= Joa dxJoy f (x,Y)dY ab dx foa f (,,,Y) dY a +fb dx fy_bf(x,y) dy. 3 Sup ...

672 Iterated and multiple integrals provided the integrals exist. With the aid of series and termwise integration (which can be ...

13.3 Double integrals 673 Be sure to realize that this is correct because, for each fixedy, Q(x,y) is a function of x whose deri ...

674 Iterated and multiple integrals in which the right member is a determinant. The expanded form of (12) is (13) VXV=(ay-a.)i+( ...

13.3 Double integrals 675 of this problem. One example is sufficientto expose an interesting idea that enables us to establish ( ...

676 Iterated and multiple integrals that V X V is continuous over the region implies that V X V = 0 over the region. On the othe ...

13.4 Applications of doubleand iterated integrals 677 tions y = 0, y = x2, and x = 1. We suppose that, at each point (x,y) of th ...

678 Iterated and multiple integrals With the aid of Theorem 13.38 we can quickly express the above double integral as an iterate ...

13.4 Applications of doubleand iterated integrals 679 A simple modification of the precedingparagraph gives an iterated integral ...

680 Iterated and multiple integrals can put these equations in the form (13.461) x ffs S(x,y) dx dy =Ifs xS(x,y) dx dy, y ffs S( ...

13.4 Applications of double and iterated integrals 681 lamina about the line through the centroid parallel to the linex = X. The ...

682 Iterated and multiple integrals Problems 13.49 1 A rectangular lamina has opposite vertices at the origin and the point (a,b ...

13.4 Applications of double and iteratedintegrals 683 10 Solve Problem 9 again, using the parametric equations to obtain integra ...

684 Iterated and multiple integrals line x = xa. Ans.: M(p) xa dx yi+m2(x-xl) x-xo ixi yi+m.a(x-xt) X3 y,+m,(x-x,) + fSa dx yx+' ...

13.4 Applications of double and iterated integrals^685 < a, and we can beinterested in situations in which e is small. Set up ...