00:01
Here we'd like to find the volume obtained by rotating the region bounded by the curves, sine and cosine, about the line y equals 1.
00:10
So here's a rough sketch of the graph down here in red.
00:14
This is the graph of sign from 0 to pi over 4.
00:20
And then in blue we have the cosine graph.
00:24
The green line is the axis of rotation.
00:28
This is the horizontal line, y equals 1.
00:31
And then the red and the blue graphs up top.
00:34
Those are the reflections after we've rotated this area down here around the line y equals 1.
00:44
So we see after we do a revolution around the line y equals 1, we have this hole in the middle of our solid.
00:52
So that means that our cross sections, which is indicated by this washer right here, will have holes, so we have washers.
01:18
So we have a formula for the volume in this case.
01:21
Also, we obtain the volume by letting the washers move in the x direction, so our volume will be in terms of x.
01:33
So we know the volume is the integral, 0 to pi over 4, times the area of the washer, which is pi times the larger radius squared minus the smaller radius squared.
01:49
So we need to find formulas for big r and little r.
01:55
So let's rely on the picture for this.
02:01
So looking at the picture here, we want a large radius, which is the distance from the center all the way out to the endpoint.
02:11
So this is big r.
02:17
So one way to get there is to take this entire distance from the x -axis to the horizontal green line, which is one.
02:30
And then if we subtract this distance right here from zero to the red graph then informally we have a quality here so we take the whole distance which is one and we subtract off the distance from x all the way up to the red and we're left over with this big r so we have big r equals the entire line which is one minus this y value here but since this y values on the red sign graph, we could replace y with sinex.
03:16
So that gives us big r.
03:32
And then for little r, the same idea.
03:36
We want this smaller line right here...