Volumes
Find the volume of the solid generated by revolv...

Question

Volumes

Find the volume of the solid generated by revolving the region bounded by the curve y = sin x and the lines x = 0, x = π and y = 2 about the line y = 2.

Accepted Answer

Hello there. Today we are going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Determine the volume of the solid formed by revolving the region bounded by the curve y equals cosine of x and the lines x is equal to 0, x is equal to pi divided by 2, and y is equal to 3 about the line y is equal to 3. OK, so it appears for this particular problem. We're asked to determine the volume of the solid that is formed by revolving the region that is bounded by the specific curve y is equal to cosine of x and the lines x is equal to 0, x is equal to pi divided by 2, and y is equal to 3, and we're revolving it around the line or about the line y is equal to 3. So essentially, we are ultimately trying to solve for the volume for this specific problem based on all the information that is provided to us. So now that we know that we're ultimately trying to solve for the volume, what we need to do first in order to solve this problem is we need to recall and use the DC method formula, which states that the volume V will be equal to. Pi multiplied by the integral evaluated from A to B of parentheses of r x to the power of 2. DX. So this is the disc method formula, and we need to recall a note that RX represents the radius of the disc. So, based on what we're told in the problem, we need to note that the region between, once again, the region between Y is equal to cosine of X and Y is equal to 3. Or 0 is less than or equal to x and x is less than or equal to pi divided by 2. So once again, here for this specific problem, the region is between y is equal to cosine of x and y is equal to 3, or 0 is less than or equal to x and x is equal to pi divided by 2. And when we revolve about Y equals 3, this gives discs, this will give us a disc who has a radius that is, once again, We need to note that we're revolving about the line Y equals 3, and this will give us a disk whose radius is the vertical distance from the Y axis. Y is equal to 3, down to the curve, Y is equal to cosine of X. So that means, so let's make a note, since this is very important, that R of X will be equal to 3 minus cosine of X. So that means our radius is equal to 3 minus cosine of X. So, therefore, we can go ahead and write that the volume V will be equal to pi multiplied by the integral evaluated from 0 to pi divided by 23 minus cosine. Of x to the power of 2, since it's 3 minus cosine of x in parentheses to the power of 2. DX which will be equal to. Pi multiplied by the integral evaluated from 0 to pi divided by 2. Of parentheses of 9 minus 6, so 9 minus 6 multiplied by cosine of x. Plus Cosine square of XDX. Fantastic. We also need to recall that cosine of 2 x is equal to 2 multiplied by cosine square x minus 1. So that means we can go ahead and write that cosine square x will be equal to 1/2 multiplied by cosine of 2 x plus 1/2. Thus, we can go ahead and write that the volume V. is equal to pi multiplied by the integral evaluated from 0 to pi divided by 29 minus 6 multiplied by cosine of x + 1/2 multiplied by cosine of 2 x plus 1/2. Dx, which will mean that v is equal to pi multiplied by the integral evaluated from 0 to pi divided by 2 of parentheses of 19 divided by 2 minus 6 multiplied by cosine of x plus 1 divided by 2 multiplied by cosine of 2 x d x. Which we can write that v will be equal to pi multiplied by square brackets of 19 divided by 2 multiplied by the interval evaluated from 0 to pi divided by 2. Of dx minus 6 multiplied by the integral evaluated from 0 to pi divided by 2 of cosine of x plus 1/2 multiplied by 1/2 multiplied by the integral evaluated from 0 to pi divided by 2 of cosine of 2 x. D, and it's gonna be cosine of 2 X multiplied by D of 2X. fantastic. And don't forget your square bracket. So that will mean we can simply say that V is equal to. Pi, and we need to take pi and we need to multiply it by square brackets of 19. Divided by 2 multiplied by x minus 6 multiplied by sin of x plus 1 divided by 4. Multiplied by sin of 2 x and we need to evaluate it at 02 I divided by 2. And we need to note, let's make a note here that at, so once again, at X is equal to pi divided by 2, we can go ahead and write that. 19 divided by 2 multiplied by pi divided by 2 minus 6 multiplied by sin of i divided by 2 plus 1 divided by 4 multiplied by sin of 2 multiplied by the parentheses of 2 of pi divided by 2. Will be equal to 19 i divided by 4 minus 6 multiplied by 1 plus 1 divided by 4 multiplied by sin of pi, which will be equal to 19 pi divided by 4 minus 6. Fantastic. And we need to note that at X is equal to 0. We can go ahead and write that when x is equal to 0, we can write that 19 divided by 2 multiplied by 0 minus 6 multiplied by sin of 0. Plus 1/4 multiplied by sins of 2 multiplied by 0 will be simply equal to 0. Thus, we can go ahead and write that V the volume will be equal to pi multiplied by square brackets of 19 pi divided by 4 minus 6, and don't forget your square bracket. So, that means that our final answer without a doubt will be that the volume V will be equal to 19 to 2. -24 multiplied by pi. All divided by 4, and don't forget that our units are going to be cubic units because we are dealing with volume. And that's it. We've officially solved for this problem. Hooray, we did it. So going back to the top to recap what we've learned and based on what we've learned, we could safely say once again that our final answer for the volume for this particular solid. Will be once again as follows, 19 pi to the power of 2 minus 24 pi divided by 4 cubic units, and that's it. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

VolumesFind the volume of the solid generated by revolving the region bounded by the curve y = sin x and the lines x = 0, x = π and y = 2 about the line y = 2.

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Volumes
Find the volume of the solid generated by revolving the region bounded by the curve y = sin x and the lines x = 0, x = π and y = 2 about the line y = 2.