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.