 In this video, we provide the solution to question number eight for practice exam number one for math 1220 We're asked to set up the integral to calculate the volume of the region bounded by the curves y equals sine of x y equals zero x equals zero and x equals pi That is rotated about the x-axis. So we're looking at a solid of revolution here. Let's first consider the region in play So if we sketch the x-axis and the y-axis like usual notice that the line Y equals zero is just the x-axis the line x equals zero is the Y-axis and then x equals pi that's going to be another vertical line in the Over here. So let's put something like this on the screen. I'll label it for good measure x equals pi And then we look at the curve Y equals sine for which sine is going to do something like the following between zero and pi It's going to go up reach its maximum of pi halves and then I'll return to an x-intercept at x equals pi so the region we're considering has got to be this region right here, okay? We want to rotate this thing around the x-axis Like so so a typical well, we have to make a decision Do we want to use the the disc method or the shell method? I think the disc method would be a little bit cleaner here Because a typical cross-section would look something like the following the thickness of this cross-section would be a dx I definitely want to integrate with respect to x because as there's a sign of x involved if I integrated respect to y I'd have to do an arc sign. Yes, that would be that would be a lot more challenging So we're going to take this and rotate it around the x-axis forming a then a disc So by the disc method the volume will equal pi times the integral of the radius squared The radius is going to be the height of this rectangle and so using this point on the curve x comma y Where are the height is going to be the y-coordinate, but we're going to integrate respect to x Since we're integrating respect to x we're going to go from the far left of this picture can go Which would be x equals zero and then the far right that these rectangles can go would be x equals pi So we'll integrate from zero to pi like in that manner I do need to get rid of the y But given the curve is y equals sine of x the height of This rectangle is given by the sine curve And so that tells us that the volume would be pi times the integral from zero to pi of Sine squared x dx remember this integral is what we have to set up We do not have to evaluate it So even though there might be some trig identities you could use to compute that thing that's not what this question As he'd only asked us to set up the integral and when we do so we end up with what's seen on the screen