Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.
$ x = -2 - y^2 $ , $ x = y^4 $ ; about the y-axis
Applications of Integration
before I begin, I want to quickly note that there's a typo on this question. In the actual textbook, we don't have this negative sign right here. It's just the line or the curve. What X equals two minus y squared in the region is also bound by executes. Whatever that fourth, we're taking around about the Y axis. So you can first begin by drawing out the region. So we No, they're usually so Why cause actual fourth looks something like this soon, actually, the X equals y to the fourth. Looks like this on Now we know that acts to the Y squared look similar to this green problem that drew. But because we're negating it with this, it looks like this. We're shifting it to the right by two. So this is what we get. And so now you get the region right here. We're rotating this about the Y axis putting washer. The smallest ripple. They're bigger. Ship will there. Now we want to find the two points of intersection and we can do this by setting the equal to each other two months. Why? Square equals. I went to the fourth. These easiest way to arrive at the solution is really Teo. First move, so two equals. Why? To the fourth plus why square, like the obvious solutions if you think about it, is that if each of these elements were one then that they would add to and therefore their questions would be solved. So on if y to the fourth is equal to one and why squared is equal to one, then why can be negative one or one And so this point we're here. It's just one negative one work one This part here is one negative one. So those are the two points of interception when you find the cross sectional area and that is a function why that is equal to larger area minus the small area. A larger area is created by this cover. Here is a constantly greater The X Files is constantly greater so you can plug in the value of this which is this curse this problem. So yeah, that's equal to pi two months. Why square squared and the smaller radius is created by this curve. There's X ingles wants the fourth, And so when we multiply this out, put the pie, we get that this is equal to pi Two Squared is four he comes to is for Why square close to the power of two is once before Oh Then we'LL stop. I don't want to close the sprints easier because I did pull up a pie at the very beginning. Why did the fourth square is one eighth? And so this is the cross sectional area function. We can put this into the interval. So as we saw before, we're going from negative one to one. No. And so we can first pull up the pie and you put in the car section area function. So that gives us for minus or why scores plus once before minus the eighth as the same thing as what we had before d'Oh And so now we can just evaluate it. That's the entire derivative. Four. Wiseman, remove off for the anti drug of four y squared is former three white cute The anti drew of y to the fourth is one to the fifth over five in the interview off. Why it's the eighth is white on ninth over nine, and all this has evaluated at native one and one. So now he's in this plugging y equals one and y equals native one and solve. Since four times, one is for once. The third is one. So it's minus four three for thirds plus one fist minus one nine. All this minus pi times. Are you planting native one? I made it for plus four thirds minus y minus one fifth. Plus you're nice. And when we do all of this out, the final answer is two four eight pie over forty five. That is our final answer.