Volumes of Revolution Question

[RJLiberator]

Homework Statement

Let R be the region bounded by the curves y=0 and y=x^2+x between x=0 and x=1. Compute the volume of the solid of revolution obtained when R is rotated about the axis y=-1.

Homework Equations

Disc method: integral of pi*r^2 = volume from the bounds

The Attempt at a Solution

The bounds are obvious = 0 to 1.

The hard part is accounting for the empty area inbetween as we rotate around y=-1 rather then y=0.

What I did to solve the problem was moved the volume up 1 to the origin. I then solved the entire volume of the problem from 0 to 1 using the radius of x^2+x+1 (the +1 to account the entire area).
My answer here was 37pi/10.

I then went back and took the volume of the squared middle region which had radius of 1 and then subtracted that from my first answer.

My final answer was 27pi/10.

Can anyone confirm my train of thought/answer?

Thank you.

 

[Mark44]

Homework Statement

Let R be the region bounded by the curves y=0 and y=x^2+x between x=0 and x=1. Compute the volume of the solid of revolution obtained when R is rotated about the axis y=-1.

Homework Equations

Disc method: integral of pi*r^2 = volume from the bounds

The Attempt at a Solution

The bounds are obvious = 0 to 1.

The hard part is accounting for the empty area inbetween as we rotate around y=-1 rather then y=0.

What I did to solve the problem was moved the volume up 1 to the origin. I then solved the entire volume of the problem from 0 to 1 using the radius of x^2+x+1 (the +1 to account the entire area).
My answer here was 37pi/10.

I then went back and took the volume of the squared middle region which had radius of 1 and then subtracted that from my first answer.

My final answer was 27pi/10.

Can anyone confirm my train of thought/answer?

Thank you.

That works for me.

You could also do the problem without translating the region upward like so:
$\Delta V = \pi[(\text{outer radius})^2 - (\text{inner radius})^2]\Delta x$
where the outer radius is $x^2 + x - (-1) = x^2 + x + 1$ and the inner radius is 1.
The volume integral is then
$\pi \int_0^1 [(x^2 + x + 1)^2 - 1^2] dx$

 

[RJLiberator]

Ah, so you managed to get the same answer as well? Confirming my result :D

 

[Mark44]

Yes, same answer.