Method of Cylindrical Shells

[simplybilly]

[FONT=verdana, helvetica, sans-serif]Edit: Ok So I finished my last problem and figured out the solution.

However I am completely lost on this next problem.

If someone could give me either a similar problem and show how to do the work or work out this problem and explain how to do this.

[/FONT]Find the volume V obtained by rotating the region bounded by the given curves about the specified axis.y = 5 sec x, y = 5 cos x, 0 ≤ x ≤

π

3

; about y = −5

 

[daon2]

What is your difficulty?

\(\displaystyle V = 2\pi\int_0^1 x(e^x-e^{-x})dx\)

 

[simplybilly]

What is your difficulty?

\(\displaystyle V = 2\pi\int_0^1 x(e^x-e^{-x})dx\)

No I discovered it. When I was submitting my answer I messed up and hit a 3 instead of an e. The answer was 4pi/e. However, I have another problem that I am actually confused about.

 

[daon2]

No I discovered it. When I was submitting my answer I messed up and hit a 3 instead of an e. The answer was 4pi/e. However, I have another problem that I am actually confused about.

Use the disk/washer method.

\(\displaystyle r_1(x) = 5\sec(x)-(-5)\)
\(\displaystyle r_2(x) = 5\cos(x)-(-5)\)

\(\displaystyle V= \pi\int_0^{\pi/3}(r_1(x)^2-r_2(x)^2)dx\)

Also, please don't keep editing your original post. It will be confusing for others to read replies to it when it changes.

 

[MarkFL]

The volume of an arbitrary shell is:

\(\displaystyle dV=2\pi rh\,dx\) where:

\(\displaystyle r=x\)

\(\displaystyle h=e^x-e^{-x}\)

and so:

\(\displaystyle dV=2\pi x\left(e^x-e^{-x} \right)\,dx\)

summing the shells, we find:

\(\displaystyle \displaystyle V=2\pi\int_0^1x\left(e^x-e^{-x} \right)\,dx\)

I would proceed with integration by parts.

edit: Nevermind...the problem has changed. Someday I will learn to quote...

 

[DrPhil]

Edit: Ok So I finished my last problem and figured out the solution.

However I am completely lost on this next problem.

If someone could give me either a similar problem and show how to do the work or work out this problem and explain how to do this.

Find the volume V obtained by rotating the region bounded by the given curves about the specified axis.y = 5 sec x, y = 5 cos x, 0 ≤ x ≤

π

3

; about y = −5

I certainly found the references to e^x confusing - bot having seen your preceding question in the same thread. New thread for new question, PLEASE!

Assuming you are to do this by cylinders (instead of washers), the increment of volume of a cylindrical shell at a value of x in (0, pi/3) is circumference times height times dx. Since the axis of rotation is at x=-5, the radius of the shell is (x + 5).

\(\displaystyle dV = 2 \pi r(x) h(x) dx\)

where \(\displaystyle r(x) = (x + 5) \) and \(\displaystyle h(x) = 5\left[\sec(x) - \cos(x)\right]\)

Integrate \(\displaystyle dV\) from \(\displaystyle x=0\) to \(\displaystyle x=\pi/3\)