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Assignment 9

# Assignment 9 - Math 237 Assignment 9 1 Use T(x y =(x y x y...

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Math 237 Assignment 9 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 R x 2 + y dV where R is the region bounded by x + y + z = 2, z = 2, x = 1 and y = x . 4. Evaluate RRR R xyz e x - y + z dV , where R 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. 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 )
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