Question

Evaluate the integral that gives the
volume of the solid formed by revolving the region about the x-axis

Answer 1

Solids of revolution? :( grr these are so fun!!
I'm super busy though, gotta get some homework done t.t

I'll try to give you a few tips to get going on this though.

So we'll use the discs/washer method.

We take a slice, spin it around the x-axis and get these for our inner and outer radii:\[\Large r\quad=\quad 2, \qquad\qquad R\quad=\quad 4-\frac{x^2}{4}\]

So the volume of our washer will be:\[\Large \bf v\quad=\quad \pi(R^2-r^2)dx\]

Adding up all the washers within a certain region will give us the total volume:\[\Large \bf V\quad=\quad \int\limits v\]

Answer 2

So I guess the other important step is to find where we're integrating `from` and `to`.
We need points of intersection, so we set the 2 functions equal to one another and solve for x.\[\Large y=2, \qquad y=4-\frac{x^2}{4}\]
\[\Large 2\quad=\quad 4-\frac{x^2}{4}\]