Volume By Cross-Sections Help

**Hyrox** · April 6th 2008, 02:58 PM

We have a project where we have to make a model of finding volume by cross-sections, and I'm somewhat confused. This is the problem:

Find the volume - base bounded by y=x+1 and y=x^2-1, cross-sections are semi-ellipses of height 2 (perpendicular to x-axis)

And I have the formula piAB/2, A=height and B=radius. I subtracted the top function minus the bottom to get the base, and divided that by 2 to get the radius, but I don't really understand the height part. Why is the height given a constant? Shouldn't it change throughout the index?

**galactus** · April 6th 2008, 03:41 PM

The region is $(x+1)-(x^{2}-1)=-x^{2}+x+2$

The area of a semi-ellipse is $\frac{{\pi}ab}{2}$ .

The height, a, represents the height at the minor axis of the ellipse. It is constant.

The radius, b, is $\frac{-x^{2}+x+2}{2}$ . That is the major axis of the ellipse. The one you are integrating over because it changes along the region.

So, we have $\frac{{\pi}(2)(\frac{-x^{2}+x+2}{2})}{2}={\pi}(\frac{-x^{2}+x+2}{2})$

$\frac{\pi}{2}\int_{-1}^{2}[-x^{2}+x+2]dx$

**Hyrox** · April 6th 2008, 03:50 PM

Well, that reassured me a bit. I did that exactly, my only problem is the height. My only fear is that we are making a 3D model of it; wouldn't that mean that the height of all the semi-ellipses would be 2 and only the width would change? So would the height remain uniform throughout my entire model? In most other equations we have done, which aren't semi-ellipses, but the height changes as a result of the base, if that makes sense. Any help there?

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