Multivariable Calculus Video Requests

Let

S

be the cylinder with height

8

and radius

3

whose axis is parallel to the

z

-axis and whose lower base is centered at the origin.

What is the triple integral of $x$ ‍ over $S$ ‍ in cylindrical coordinates?

Answer:

First, what are cylindrical coordinates?

\begin{aligned} x & = r \cos (θ) \\ y & = r \sin (θ) \\ z & = z \end{aligned}

Second, we need to find bounds. We see that the bottom of the cylinder rests at

z = 0

, and it has height

8

, so the top is at

z = 8

. Thus,

0 < z < 8

. The radius is

3

, so

0 < r < 3

. Finally, we are asked to integrate over the whole cylinder, so

0 < θ < 2 π

\int_{0}^{8} \int_{0}^{3} \int_{0}^{2 π} \dots d θ d r d z

Third, we need to convert the integrand into cylindrical coordinates. Here,

x = r \cos (θ)

\int_{0}^{8} \int_{0}^{3} \int_{0}^{2 π} r \cos (θ) \dots d θ d r d z

Finally, we need to multiply by the Jacobian of cylindrical coordinates, which is just

r

The final answer is:

\int_{0}^{8} \int_{0}^{3} \int_{0}^{2 π} r^{2} \cos (θ) d θ d r d z

(You can evaluate this in the video if you want.)

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