Find the volume of the region bounded by z = 0,x2 +
(y − 2)
2 = 4, and
z =
∘-------
x2 + y2
.
PICT
Find the volume of the region z ≥ 0,x2 + y2≤ 4, and z ≤ 4 −
∘ -2---2-
x + y
.
Find the volume of the region which is between the surfaces z = 5y2 + 9x2 and
z = 9 − 4y2.
PICT
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.
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.
Find the volume between z = 3 − x2− y2 and z = 2
∘ --------
(x2 + y2)
.
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?
Find the volume of the cone defined by z ∈
[0,4]
having angle π∕2. Use spherical
coordinates.
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.
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?
In Example 25.3.10 on Page 1546 check out all the details by working the integrals
to be sure the steps are right.
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.
Convert the following integrals into integrals involving cylindrical coordinates and
then evaluate them.
∫−22∫0
√4−-x2
∫0xxydzdydx
∫−11∫−
√----
1−y2
√1−-y2-
∫0x+ydzdxdy
∫01∫0
√ ----
1−x2
∫x1dzdydx
For a > 0,∫−aa∫−
√-2--2
a −x
√a2−x2-
∫−
√ --------
a2− x2−y2
--------
√ a2−x2−y2
dzdydx
∫−11∫−
----
√1−x2
√1−x2
∫−
-------
√4−x2−y2
√4−x2−y2
dzdydx
Convert the following integrals into integrals involving spherical coordinates and
then evaluate them.