MATH V1207 Midterm: MATH V1205 Columbia Fall01Mid1Ans
31 Jan 2019
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Solutions to first midterm exam: oct. 4, 2001: the ice cream cone is given in spherical coordinates by 0 3, 0 0, 0 2 where 0 is the angle in the rst quadrant whose cotangent is 1/2. The density written in spherical coordinates is k . Thus the spherical integral computing the total mass is given by. Since cot( 0) = 1/2, it follows that cos( 0) = 1/ 5 and hence we have. 5: interchanging the order of integration yields. We subsitute u = x4 + 1 so that du = 4x3. X = a/c and z: we view the plane p as the graph of the function z = (2 ax by)/c. The region of the plane in the rst quadrant is a triangle with vertices ( 2 c ). Its projection d to the xy-plane is the triangle with vertices (0, 0), ( 2.
