### Video Transcript

Consider the region bounded by the curves ๐ฆ equals ๐ฅ cubed and ๐ฆ equals ๐ฅ for ๐ฅ is greater than or equal to zero. Find the volume of the solid obtained by rotating this region about the ๐ฅ-axis.

Now to help us actually visualize whatโs going on, what Iโve done is actually drawn a sketch of the graphs weโve got. So weโve got ๐ฆ equals ๐ฅ cubed and ๐ฆ equals ๐ฅ. And Iโve also only sketched the first quadrant, and thatโs because weโre only interested in the values of ๐ฅ greater than or equal to zero as it says in the question. So now what weโre looking to find is actually the volume of the solid obtained by rotating our region that weโve got here about the ๐ฅ-axis.

To help us to understand actually how weโre gonna do this, Iโm just gonna look at a little small region. So Iโve got this region here. Now if we actually rotate this round, we get this shape, which looks a little bit like a washer. So to start off with, weโre gonna have a look at it as a 2D shape because we want to find actually its surface area. So how to find the area of this shape? Well the area of the shape would actually be calculated by finding the area of the large circle, which Iโve put here with capital ๐
being its radius, minus the area of the small circle or the missing section middle. And Iโve used the radius here as small ๐.

So therefore, we can think of the area as ๐๐
squared, so ๐ capital ๐
squared, minus ๐ little ๐ squared. Well if we take ๐ out as a factor, itโd be ๐ multiplied by ๐
squared minus little ๐ squared. Okay, so thatโs the area. But okay thatโs useful, but we want to actually find the volume. So what do we need to do now. Well if we can actually consider this small section weโre looking at, well we rotate it around the ๐ฅ-axis. And when we do this, we actually have this shape which we mentioned already. But if we said it actually had a width or depth, then this actual depth would be known as ๐๐ฅ, so a very very small change in ๐ฅ because this is only a very very small portion of our region.

So therefore, what we could say is well our volume is going to be equal to ๐ and then big ๐
squared minus little ๐ squared, and thatโs because that was our surface area, then multiplied by our depth, which is ๐๐ฅ. However, this is very useful and actually helps us. But still, itโs not what weโre looking for because this is just the volume of a small section. What weโre actually looking for is the volume of the whole region, which could become like a cone with its center taken out. So how can we actually find out the volume of the whole region?

So actually if we want to find the total volume, thatโs when bring it into definite integral because the total volume is equal to the definite integral with our limits ๐ and ๐ of ๐ multiplied by capital ๐
squared minus ๐ squared ๐๐ฅ. And the reason this actually works is because what the definite integral does is it says that actually weโve got an infinite number of our small little sections between the two limits that weโve set, and it add them all together to find the volume. So the reason I say an infinite number is because if actually it was a definite number of sections then it only be an estimate. Because itโs the definite integral, what it does is actually find an infinite number of sections. And thatโs why we get our volume.

But this is kind of useful, but what does it mean, capital ๐
and little ๐? How can we actually use this to solve a problem? Well actually in general so this can help us to solve a problem like this, what we say is the definite integral between ๐ and ๐, so our limits, of ๐ and then multiplied by ๐ of ๐ฅ squared, so one function squared, minus ๐ of ๐ฅ squared ๐๐ฅ. Well to help us understand how this would work with the question that weโre looking at, well our small ๐ is gonna be equal to ๐ฅ cubed. So thatโs going to be our ๐ of ๐ฅ squared. And thatโs because we can see that actually the inner circle will actually touch the graph ๐ฆ equals ๐ฅ cubed first.

And our capital ๐
would be equal to ๐ฅ. And thatโs because the radius of the larger circle actually touches the line ๐ฆ equals ๐ฅ. So in the formula, this would be actually ๐ of ๐ฅ. Okay, so now what weโve done is weโve actually found the formula, and weโve actually seen where it comes from. So letโs get on and solve the problem. The first thing we need to do is actually think about what our bounds are going to be. So this is our ๐ and our ๐.

Well weโre gonna be able to find them in a second, but we also know that the ๐, our lower bound is zero, because we can actually see that actually both lines go to the point zero, zero. Now in order to actually calculate what our bound is going to be, what we need to do is make each of our functions equal to each other. So weโve got ๐ฅ cubed is equal ๐ฅ. So now Iโll actually subtract ๐ฅ from each side of the equation, and weโre gonna get ๐ฅ cubed minus ๐ฅ is equal to zero. And then if I take out ๐ฅ as a factor, Iโm gonna have ๐ฅ multiplied by ๐ฅ squared minus one is equal to zero.

And then we can actually factor one stage further. And thatโs because we know that ๐ฅ squared minus one is a difference of two squares. So now fully factored, weโve got ๐ฅ multiplied by ๐ฅ plus one multiplied by ๐ฅ minus one is equal to zero. So therefore, the points where the two functions actually meet will be ๐ฅ equals zero, negative one, and one. So now weโve got the limits zero and one. And weโve got those because well we already knew it is zero because we found it earlier and one because that is gonna be our upper limit because we can actually disregard negative one because ๐ฅ needs to be greater than or equal to zero.

So therefore with our limits substituted in for ๐ and ๐, weโre gonna say that the volume is equal to the definite integral with limits one and zero of ๐ multiplied by then weโve got ๐ฅ squared minus ๐ฅ cubed all squared ๐๐ฅ. And thatโs because we knew which way round to put it because we said that our capital ๐
was going to be our ๐ฅ. So in this case in our integral formula, itโs gonna be ๐ of ๐ฅ and then our ๐ฅ cubed was actually going to be our small ๐ or our ๐ of ๐ฅ in our formula. So therefore, if we actually simplify this, weโre gonna have the volume is equal to ๐, because we can take it out because itโs just a constant, multiplied by the integral between limits one and zero of ๐ฅ squared minus ๐ฅ to the power of six. And we got that cause itโs ๐ฅ cubed squared cause itโs ๐ฅ to the power of six.

So therefore, this is gonna be equal to ๐ multiplied by then weโve got ๐ฅ cubed over three minus ๐ฅ to the power of seven over seven, with the limits one and zero. Just to remind us how we did that, well actually if we going to find the integral of our expression, then weโll pick this term here. So first of all, you raise the exponent by one so we went from six to seven because you raised the exponent by one. And then you divide by the new exponent, so seven.

Okay, great! Weโve done that. Whatโs the next step? Well to actually find the value of a definite integral, what we can say is that the value of a definite integral between the limits ๐ and ๐ of a function is equal to the integral of that function with ๐ substituted in for our ๐ฅ minus the integral of that function with ๐, our lower limit, substituted in for ๐ฅ. So therefore, what weโre gonna get is ๐ multiplied by and then one cubed over three minus one to the power of seven over seven minus zero cubed over three minus zero to the power of seven over seven.

I wouldnโt usually put the zeros in. I just say that it is equal to zero. But I just want to show how the process actually works. So therefore, the volume is gonna be equal to ๐ multiplied by a third minus a seventh. So what we do now is actually find a common denominator. So our common denominator is going to be 21. So therefore, weโve got ๐ multiplied by seven over 21 minus three over 21. So we can say therefore that the volume of the solid obtained by rotating a region bounded by the curves ๐ฆ equals ๐ฅ cubed and ๐ฆ equals ๐ฅ for ๐ฅ being greater than or equal to zero about the ๐ฅ-axis is going to be equal to four over 21๐.