Volume of a Solid By Revolution

[Justabeginner]

Homework Statement

Find the volume of the solid generated by revolving about the line x=-1, the region bounded by the curves y=-x^2 + 4x - 3, and y=0.

Homework Equations

Shell Method?

The Attempt at a Solution

V= 2pi * ∫x* f(x) dx, where a and b are the lower and upper limits of integration, respectively.
I'm not even sure if what I'm doing is right. And how do I know to use dx or dy? Guidance please?

 

[mfb]

Shell Method?

Okay

V= 2pi * ∫x* f(x) dx, where a and b are the lower and upper limits of integration, respectively.

What does the single x there represent in your shells?
Hint: if you rotate around x=-1 instead of x=0, you have to change this.

And how do I know to use dx or dy?

Do you have constant x or constant y within your shells? That is fine.

 

[Justabeginner]

The x represents the radius, and so I would have to add 1 (- -1)? But I don't even understand what quantity defines the radius here. Is it the -x^2+4x-3? There is no constant given, but I presumed I should use dx since the problem is rotating around the x=-1 line (shift of x-axis left 1 unit).

 

[mfb]

and so I would have to add 1

Right.

But I don't even understand what quantity defines the radius here.

The integration variable ("dx") determines that radius.
The function value (-x^2+4x-3) is the "height" of the shell.

 

[Justabeginner]

So I would end up with

2 pi * integral sign [(x+1) (-x^2 + 4x-3) dx] ? Is that really it? o.0 I thought it's more intricate.

 

[mfb]

Most problems are easy, if you know how to solve them.

 

[Justabeginner]

Wow, thank you so much mfb. I appreciate it.