a11 - Math 237 Assignment 11 1 Use T(x y =(x y x y to...

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Math 237 Assignment 11 Not to be handed in 1. Use T ( x,y ) = ( x + y, - x + y ) to evaluate R π 0 R π - y 0 ( x + y ) cos( x - y ) dx dy . 2. Find a linear transformation that maps x 2 + 4 xy + 5 y 2 = 4 onto a unit circle. Hence show that the area enclosed by the ellipse equals 4 π . 3. Evaluate RRR D x 2 + y dV where D is the region bounded by x + y + z = 2, z = 2, x = 1 and y = x . 4. Evaluate RRR D xyz e x - y + z dV , where D xyz is bounded by the planes x - y + z = 2, x - y + z = 3, x + 2 y = - 1, x + 2 y = 1, x - z = 0 and x - z = 2. 5. Let 0 < a < b and 0 < c < d . Show that the region D in the first quadrant bounded by ay = x 3 , by = x 3 , cx = y 3 and dx = y 3 has area 1 2 ( b - a )( d - c ). 6. Evaluate the following integrals. a) RRR D ( x 2 + y 2 + z 2 ) - 3 / 2 dV where D is the bounded by x 2 + y 2 + z 2 = a 2 and x 2 + y 2 + z 2 = b 2 with 0 < b < a . b) Z 1 0 Z 1 - x 2 0 Z 2 - x 2 - y 2 x 2 + y 2 dz dy dx . c) Z 1 0 Z 1 - y 2 0 Z 3 3 x 2 +3 y 2 dz dx dy . 7. Find the volume of the region bounded by the surfaces.
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