(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

The area enclosed between the ellipse [itex]4x^2 + 9y^2 = 36[/itex] and its auxiliary circle [itex]x^2 + y^2 = 9[/itex] is rotated about the y-axis through [itex]\pi[/itex] radians. Find, by integration, the volume generated.

This is the whole question. I assume it means bounded by the x-axis, but even if this isn't the case, my answer is wrong. :(

Solution:[itex]12\pi[/itex]

2. Relevant equations

[tex]V = \int_a^b{A(y)\,dy}[/tex]

3. The attempt at a solution

[tex]x^2 + y^2 = 9 \Rightarrow x^2 = 9 - y^2[/tex]

[tex]4x^2 + 9y^2 = 36 \Rightarrow x^2 = \frac{36-9y^2}{4}[/tex]

[tex]A(y) = \pi \left [ \left ( 9-y^2 \right ) - \left ( \frac{36-9y^2}{4} \right ) \right ] = \frac{5\pi}{4}y^2[/tex]

[tex]\int_0^3{\frac{5\pi}{4}y^2\,dy} = \frac{5\pi}{4} \left [ \frac{y^3}{3} \right ]^3_0 = \frac{45\pi}{4}[/tex]

I hope someone can help me find where I went wrong! Thanks very much.

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