00:01
In this question, we are getting a solid of revolution, and we're going to be revolving around the line y equals minus 2.
00:09
So let's try and draw this region out.
00:12
Just try and make these axes fairly straight.
00:17
X, y.
00:18
We've got the line y equals minus 2, and that's the line about which we are revolving.
00:25
And then we have the region enclosed by the squared.
00:42
So here's our region in here.
00:44
Now where do they intersect? what are the two intersection points? that's what we need to find first.
00:51
So that's when both values of y are the same.
00:53
So x squared equals 2 thirds x, or x squared minus 2 thirds x equals zero.
01:02
And that tells us that x times x minus 2 thirds equals zero.
01:07
So this gives us our lower limit, x1, which is zero.
01:11
And this gives us our upper limit, x2 equals 2 thirds.
01:19
So this is 2 thirds here, and this is zero.
01:24
Right.
01:25
So now we are going to look at the inner and outer radii of a washer formed at a particular x.
01:33
So let's pick an x.
01:37
Then this is the inner surface, is a distance this far from the line.
01:44
So this is the inner radius, and this is the outer radius.
01:50
So what are the inner and outer radii? well, the inner radius is the y value, y of x, but for x squared.
02:00
So it's going to be x squared, but we need to add on the distance from zero to minus 2.
02:04
So it's x squared plus 2.
02:07
The outer is given by 2 thirds x plus 2.
02:13
So we're adding on the plus 2 to get from the line that we're revolving around, which is here, up to zero.
02:23
And then we're adding y of x to get to our inner and outer radii.
02:30
Now part c, we are going to have a look at the volume of a slice.
02:38
So let's say that the volume of a slice is a of x times delta x...