Question

Consider the following: y = 6 cos^4(x), y = -6 cos^4(x), -π/2 <= x <= π/2; about x = π Set up an integral for the volume V of the solid obtained by rotating the region bounded by the given curve about the specified axis.

          Consider the following:
y = 6 cos^4(x), y = -6 cos^4(x), -π/2 <= x <= π/2; about x = π
Set up an integral for the volume V of the solid obtained by rotating the region bounded by the given curve about the specified axis.

Added by Ana L.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Jacob Fry

02/28/2023

Step 1

The given curve is y = 6 cos^4(x) and y = -6 cos^4(x) for -π/2 <= x <= π/2. Since we are rotating the region about x = π, we need to shift the bounds of integration accordingly. The new bounds of integration will be from 0 to π. Show more…

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Consider the following: y = 6 cos^4(x), y = -6 cos^4(x), -π/2 <= x <= π/2; about x = π Set up an integral for the volume V of the solid obtained by rotating the region bounded by the given curve about the specified axis.

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Transcript

00:01 So this question we have a solid bounded by some curves.

00:08 So here's x, y.

00:10 We have the line y equals zero.

00:13 So this is the x -axis.

00:15 We have the line y equals cos of 6x.

00:19 So that's going to look like this.

00:26 So if this is cos of 6x, then this is going to go to zero when 6x is 5x is when 6x is 5.

00:35 Is pi divided by 12.

00:38 And funnily enough, the line x equals pi divided by 12 is also a boundary.

00:44 And so is the line x equals zero.

00:47 So that is the y -axis.

00:51 And we want to rotate this around the axis.

00:54 So this is our region in here.

00:55 This is our region.

00:59 And we will rotate this around the line.

01:01 So let's just actually get rid of this.

01:07 So this is 1.

01:09 This line is cos 6x.

01:12 And we want to rotate this around the axis x equals 8.

01:20 So what is the volume of revolution? well, the volume is the double integral over the area of the region r of 2 pi times the distance from the line x equals 8.

01:33 So that's going to be 8 minus x, dx, dy...

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