Math 265: Lesson 17 Assignment Z 2 √ 4−x2 Z (x2 + y 2 ) dy dx and evaluate the inte- (1) Sketch the region of integration for 0 gral by changing to polar coordinates. 0 (2) Integrate x2 + y 2 + z 2 over the cylinder x2 + y 2 ≤ 2, 2 ≤ z ≤ 3. (3) Use cylindrical coordinates to compute the integral of f (x, y, z) = x2 + y 2 over the solid below the plane z = 4 inside the paraboloid z = x2 + y 2 . ZZZ 3 2 2 2 (4) Use spherical coordinates to evaluate e(x +y +z ) 2 dV where U is the solid unit sphere given by x2 + y 2 + z 2 ≤ 1. U (5) Use spherical coordinates to compute the volume of the solid that lies above the cone p p z = x2 + y 2 and below the hemisphere z = 1 − x2 − y 2 . (6) (Bonus) Use cylindrical coordinates to calculate the volume above the xy-plane outside the cone z 2 = x2 + y 2 and inside the cylinder x2 + y 2 = 4. Each problem is worth 10 points. The bonus is worth up to 5 points.