Calculus of One and Several Variables

25.4 Exercises

1. Find the volume of the region bounded by z = 0,x² +
   (y − 2)
   ² = 4, and z =
   ∘------- x2 + y2
   .
   PICT

2. Find the volume of the region z ≥ 0,x² + y² ≤ 4, and z ≤ 4 −
   ∘ -2---2- x + y
   .
3. Find the volume of the region which is between the surfaces z = 5y² + 9x² and z = 9 − 4y².
   PICT

4. Find the volume of the region which is between z = x² + y² and z = 5 − 4x. Hint: You might want to change variables at some point.
5. The ice cream in a sugar cone is described in spherical coordinates by ρ ∈
   [0,10]
   ,ϕ ∈
   [0, 13π]
   ,θ ∈
   [0,2π]
   . If the units are in centimeters, find the total volume in cubic centimeters of this ice cream.
6. Find the volume between z = 3 − x² − y² and z = 2
   ∘ -------- (x2 + y2)
   .
7. A ball of radius 3 is placed in a drill press and a hole of radius 2 is drilled out with the center of the hole a diameter of the ball. What is the volume of the material which remains?
8. Find the volume of the cone defined by z ∈
   [0,4]
   having angle π∕2. Use spherical coordinates.
9. A ball of radius 9 has density equal to
   ∘ ----------- x2 + y2 + z2
   in rectangular coordinates. The top of this ball is sliced off by a plane of the form z = 2. Write integrals for the mass of what is left. In spherical coordinates and in cylindrical coordinates.
10. A ball of radius 4 has a cone taken out of the top which has an angle of π∕2 and then a cone taken out of the bottom which has an angle of π∕3. Then a slice, θ ∈
    [0,π∕4]
    is removed. What is the volume of what is left?
11. In Example 25.3.10 on Page 1546 check out all the details by working the integrals to be sure the steps are right.
12. What if the hollow sphere in Example 25.3.10 were in two dimensions and everything, including Newton’s law still held? Would similar conclusions hold? Explain.
13. Convert the following integrals into integrals involving cylindrical coordinates and then evaluate them.
    1. ∫ ₋₂² ∫ ₀
       √4−-x2
       ∫ ₀^xxydzdydx
    2. ∫ ₋₁¹ ∫ −
       √---- 1−y2
       √1−-y2-
       ∫ ₀^x+ydzdxdy
    3. ∫ ₀¹ ∫ ₀
       √ ---- 1−x2
       ∫ _x¹dzdydx
    4. For a > 0,∫ _−a^a ∫ −
       √-2--2 a −x
       √a2−x2-
       ∫ −
       √ -------- a2− x2−y2
       -------- √ a2−x2−y2
       dzdydx
    5. ∫ ₋₁¹ ∫ −
       ---- √1−x2
       √1−x2
       ∫ −
       ------- √4−x2−y2
       √4−x2−y2
       dzdydx
14. Convert the following integrals into integrals involving spherical coordinates and then evaluate them.
    1. ∫ _−a^a ∫ −
       √----- a2−x2
       √a2−-x2
       ∫ −
       √ -------- a2−x2− y2
       √a2−-x2−y2-
       dzdydx
    2. ∫ ₋₁¹ ∫ ₀
       √ ---2 1− x
       ∫ −
       √1-−x2−y2
       √ ---2--2 1−x −y
       dzdydx
    3. ∫ −
       √2
       √- 2
       ∫ −
       √2−x2-
       √---- 2−x2
       ∫
       √ 2--2- x+y
       √ ------- 4− x2−y2
       dzdydx
    4. ∫ −
       √- 3
       √3
       ∫ −
       √ ---- 3−x2
       √3−x2
       ∫ ₁
       √4−x2−y2
       dzdydx
    5. ∫ ₋₁¹ ∫ −
       √1−x2
       √---2 1−x
       ∫ −
       √4−x2−y2
       √---2--2 4−x −y
       dzdydx