Advanced Calculus fi..

242 Advanced Calculus, Fifth Edition1f5. Verify that the transformation u = ex cos y, v = ex sin y defines a one-to-one mappingof the rectangle R,,: 0 9 x 5 1,0 5 y 5 n/2 onto a region of the uv-plane and expressas an iterated integral in u, v the integralR,,6. Verify that the transformatione2,S S I + e4, COS~ y sin2 y dxdy.defines a one-to-one mapping of the square 0 I x 5 1,0 5 y 5 1 onto a region of theuv-plane. Express the integral11 ;/r4 - 6x2y2 + y4dxdy . ?IR,,over the square as an iterated integral in u and v.7. Transform the integrals given, using the substitutions indicated:a) log(1 + x2 + y2)dy dx, x = u + v, y = u - v idb) {dj;_fxx d+dy dx, x = U, y = u + v8. Let R,, be the square 0 5 u 5 1,05 v 5 1. Show that the given equations define aone-to-one mapping of Ruu onto R,, and graph R,,:a) x = u +u2, y = eU b) x = ueU, y = eUC) x=2u-v2, y=v+uvi d) x = 5u - u2 + v2, y = 5v + lOuv9. Verify the correctness of (4.67) as a special case of (4.66). Show the geometric meaningof the volume element r Ar A0 Az.10. Verify the correctness of (4.68) as a special case of (4.66). Show the geometric meaniqgof the volume element p2 sin @ApA@A8.11. Transform to cylindrical coordinates but do not evaluate:a) IDR,,, x2y dx dy dz, where Rxyz is the region x2 + y2 _( 1,0 I z 5 1;1 I+x+y 2b) lolo lo (X - y2)dzdy dx.12. Transform to spherical coordinates but do not evaluate:a) f&,, x2y dx dy dz, where R,,, is the sphere: x2 + y2 + z2 5 a2;.IF7 1b) I!, Im(x2 + Y' + z2)dzdy dx.4.7 ARC LENGTH AND SURFACE AREAIn elementary calculus it is shown that a curve y = f (x), a 5 x 5 b, has lengthI13and that if the curve is given parametrically by equations x = x(t), y = y(t) fort, 5 r 5 t2, then it has length4