Solving Volume using integration

[CalculusSeeker]

Hi everyone, I'm a college student and our instructor asked to find volume using integration.
Here is the problem:
1. Find the volumed formed by revolving the ellipse x^2/a^2 + y^2/b^2 =1 about the line y=b.
The answer for this problem is 2pi^2ab^2 but what im getting is 2/3 pi^2ab^2.
I've included my solution here, can anyone tell me where i went wrong? Thank you very much for those who will answer.

 

[tkhunny]

1) If you HAVE to use the integration, did you do anything with the bottom half of the ellipse?

2) Personally, I'd listen to Pappus.

Area = [math]\pi a b[/math]
Volume Requested = Area * [math]2\pi b[/math]

 

[Steven G]

I always listen to Pappus.

It is very very good that you drew the strip that you wanted to revolve, but why did you stop at the x-axis? The strip does go a bit further that that!

 

[CalculusSeeker]

Thank you for correcting me. I'm looking for the volume of the whole ellipse and not just the first quadrant therefore the strip must go all the way down. Ok, I get it now. Thank you again.?

 

[CalculusSeeker]

1) If you HAVE to use the integration, did you do anything with the bottom half of the ellipse?

2) Personally, I'd listen to Pappus.

Area = [math]\pi a b[/math]
Volume Requested = Area * [math]2\pi b[/math]

Thank you for the reminders and everything. I'll try to solve it again.?