Question

The region W is the cone shown below. The angle at the vertex is π/3, and the top is flat and at a height of 1

3
​
. Write the limits of integration for ∫
W
​
dV in the following coordinates (do not reduce the domain of integration by taking advantage of symmetry): (a) Cartesian: (b) Cylindrical:
With a=,b=
c=,d=,
e=, and f=
Volume =∫
a
b
​
∫
c
d
​
∫
e
f
​
ddd
​
(c) Spherical:
With a=,,b=
c=,,,
e=, and f=
Volume =∫
a
b
​
∫
c
d
​
∫
e
f
​
dddd
​

Answer 1

The limits of integration for the volume integral in different coordinate systems are calculated.

Step-by-step explanation:

(a) In Cartesian coordinates, the limits of integration for the volume integral ∫WdV are:

a = 0, b = 1/3, c = -r, d = r, e = -r, f = r

(b) In Cylindrical coordinates, the limits of integration for the volume integral ∫WdV are:

a = 0, b = 1/3, c = 0, d = 2π, e = 0, f = √(r2 - z2)

(c) In Spherical coordinates, the limits of integration for the volume integral ∫WdV are:

a = 0, b = 1/3, c = 0, d = 2π, e = 0, f = π/3