prac exam 2B solutions

3 3 6 a py ex z qx pz ex y rx and qz ex 2y

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Unformatted text preview: − (x + y − 1)2 ) dxdy = = (2x−y )2 +(x+y −1)2 <4 2π 2 = 0 0 u2 + v 2 < 4 1 (4 − r2 ) rdrdθ = 3 2π 0 2 22 1 r − r4 3 12 2π dθ = 0 0 4 8π dθ = . 3 3 6. a) Py = ex z = Qx ; Pz = ex y = Rx ; and Qz = ex + 2y = Ry . Thus F is conservative. b) We want: fx = ex yz , fy = ex z + 2yz , fz = ex y + y 2 + 1. Integrating fx we get: f (x, y, z ) = ex yz + g (y, z ). Differentiating with respect to y and comparing, we get: fy = ex z + gy = ex z + 2yz . Thus gy = 2yz , which yields: g (y, z ) = y 2 z + h(z ). Plugging bac...
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