Volume - Integration Question Help

Question

So basically, with this question I have answered parts a) and b), but I am stuck in relation to part c). I am not exactly sure what the borders of my integration should be, but I figured with the given condition substitute in and simply. That's pretty much all I got for the last part. Can anyone enlighten me as to how to got about finding the volume of the solid.

**Note: the answer that is expected to be reached is V = (37*R^3)/18 - (5*pi*R^3)/24

Answer 1

For what value of $x$ will the points $Q$ and $M$ coincide at point $F$? This is the lower limit of integration. Clearly, this occurs if $LQ = LM$; i.e., $$R^2 - x^2 = (R+x)^2 \tan^2 \theta.$$ Solve this equation for $x$, and you will get two solutions, one of which is extraneous since it is negative.

The upper limit of integration is trivial: it is simply $x = R$, corresponding to the case where point $Q$ coincides with $D$.

Now the volume of the base is simply given by the expression $$\int_{x = \mathcal L}^R \left( 4(R+x)^2 \tan^2 \theta - \pi (R^2 - x^2) \right) \, dx,$$ where $\mathcal L$ is the lower limit of integration you obtained above.

Answer 2

To find the lower limit of your integral you need to find the $x$ value for the line EF. The upper limit of your integral is $x=R$.

Let T be the intersection point of EF and OD. Angle EOD = $2 \theta$ by circle theorems.

Distance OT = $x = R cos(2 \theta)$