Find the volume obtained by rotating the region bounded by the given curves about the specified

Find the volume obtained by rotating the region bounded by...

Key Concepts

Solids of Revolution

This concept involves creating a three-dimensional object by rotating a two-dimensional region around an axis. The resulting solid has a volume that can be computed using calculus techniques, typically by integrating infinitesimal volume elements that make up the entire solid.

Washer Method

The washer method is a technique used to compute the volume of a solid of revolution when the region being rotated has a gap or hole. It involves slicing the solid perpendicular to the axis of rotation to obtain washers (disks with holes). The volume is calculated by integrating the difference of the squares of the outer radius and the inner radius, multiplied by ?, over the interval of interest.

Integration Bounds and Slicing

An essential step in volume calculation is determining the correct bounds for integration, which represent the limits over which the region exists. This method typically involves slicing the region into thin sections (washers or shells), then summing up their contributions using definite integrals to cover the entire region.

Axis Translation

When the axis of rotation is not a coordinate axis, adjustments have to be made to the expression of the radii in the volume integral. This translation accounts for the distance between the original region and the new axis of rotation, ensuring that the correct distances (radii) are used when applying the washer or shell methods.

Transcript

00:02 So for this question, we're finding the volume when the area between sign and coast is rotated about the line y equals 1.

00:13 So remember that basically the volume in general is equal to the integral for the given bounds of the cross -sectional area.

00:31 Let's call that a with respect to x.

00:37 So what we're going to want to look at is what our cross -sectional area would be.

00:41 And for the record, that would be, we'd be using the washer method for this because we have two different functions like this, and neither of them is y -equal zero or anything like that.

00:53 So you don't necessarily have to do this, but it might be helpful to draw out what it would look like.

01:00 So i'm going to do that.

01:03 Basically, we're going to draw out the graph for sine and cos and y -equals 1.

01:10 So let's say this is, we're going to sketch it out.

01:15 It doesn't have to be super neat.

01:16 But yeah, and let's say this is pi over 4.

01:21 Now, basically, kos, it starts at 1, and it ends at, sorry, and it ends at 1 over 2 and cose, i mean, sorry, sign, that starts at 0 and for pi over 4, it's going to end at 1 over 2.

01:54 And finally, we're going to have our line that it's rotated about, which is y equals 1.

02:01 And now we're going to look at what our areas would be.

02:06 So to do this.

02:07 That, we're just going to need our radius for each circle, basically, each disk.

02:14 And for the green, the sign, so that's sine x.

02:23 That's going to be like this length.

02:27 And that would be, basically, if you look at it, well, at the top we have one, and then we're going to subtract the y value, which is sine x.

02:38 So that's what one of the r's is going to be, like, radius.

02:42 Radius, radii.

02:43 Now we're going to look at the other part, and that's going to be the radius of like a disk for kos.

02:51 And that's going to be similar.

02:53 It's going to accept the y value is going to be kos x.

02:56 So basically it's going to be 1 minus kos x.

02:58 That's going to be the other radius.

03:01 And you can see right from this diagram, basically since the line is above both of them, both the curves, and sinex is the one that's like lower.

03:13 It's going to have the bigger radius so that's going to be the outer ring sort of so um i'm just going to you can uh name them however you choose but i'm going to call this r o for outer and r i for our i for our inner now what we can do basically is um if we take think about the cross -sectional area then ours is going to be of course um from zero to pi over four it's going to be let me just move this down.

03:49 It's going to be our area of our outer circle.

03:54 So of course area of a circle is pi r squared.

03:58 So basically it's going to be pi r o squared.

04:01 And then we're going to subtract.

04:02 And then we're going to take our inner circles area, which is going to be pi r i squared.

04:10 And then we're going to have dx.

04:13 And i wrote what r o and r i are.

04:16 Those are going to be.

04:24 So we're going to have 1 minus sine x all squared.

04:28 And then we're going to subtract, and we're going to have pi times 1 minus kosex squared.

04:41 Okay, so there's that.

04:43 Now it's just a matter of finding this integral.