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Calculus/Calculus, Volumes of solids

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Expert: - 10/13/2009


Question
Hi, I am having trouble with this Calculus 2 question about volumes of solids. Mainly I'm having problems drawing out the solid and understanding what the cross sections look like/where they would be. Also what do I do when I have the cross-section? This is the question: Find the volume of the solid whose base is the triangle bounded by the x-axis, y-axis, and the line 2x + y = 2, and the cross-section perpendicular to the y-axis are semicircles.  

Any help you can provide would be greatly appreciated! Thanks.

Answer
Since the semicircles are perpendicular to the y-axis,
lets take the line as y = 2 - 2x.

We know that the area of a cirle is πr².
This means the area of a semi-circle is πr²/2.

We then need to integrate this area of semi-circles from 0 out to 2.

The problem becomes ∫πy²/2 dy, from y=0 to y=2.

You should be able to do the rest.

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