 Let's take another look at an example of finding the volume of solver of evolution using the shell method In this situation though our axis is not going to the x-axis of the y-axis In fact, we're going to pick a vertical axis x equals 2 and it differs from the y-axis right here Let's take the region the region given by the parabola y equals x minus x squared and we want to Integrate that region below that parabola but above the x-axis around the line x equals 2 Now if we're going to use the shell method our cross-section should be parallel to the axis So you see this is the type of cross-section we would want the thickness is going to be dx We're going to integrate with respect to x because that's how thick this thing is and so let's go through the shell method and See what we need the volume of a shell remember It's 2 pi times our radius times the height times the thickness Which is a delta R? All right, we've already taken care of delta R. We're going to integrate with respect to x So this is going to be a delta x your radius Well, let's actually let's do the height next the height is how tall is that how tall is this? Rectangle going to be the rectangles from it goes from this point on the function down to the x-axis The height is just going to be the y-coordinate of this point right here x comma y And as this is function is this this is a function the y-coordinate will be given as x minus x squared Like so and so that's what the height of this thing is going to be x minus x squared So what's left to identify is what's the radius now? The radius is going to be the distance from your cross-section to the axis right here now We know the total distance between from the y-axis to this axis. It's going to be 2 All right, we also know the distance from the y-axis to this cross-section right here It's x that's what the x-coordinate represents So what we need to figure out this right here is the radius We're looking for r and so we want to take the difference of those two values 2 minus x And now we'll give us the radius of This rotation here and once you have all the pieces the radius the height and the thickness We're then ready to set up our integral The volume is going to equal 2 pi the integral of your radius which we got us 2 minus x the height Which was x minus x squared and then the thickness which is going to be a dx right here Now we have to still determine. What are the bounds of integration? We're integrating with respect to x so we're going to be looking for x coordinates x equals 0 x equals z x equals something up to x equals something I gave one away there the left most fx coordinates going to be x equals 0 this point Right here so that goes on the lower bound and what's the right most coordinate? Well, it did tell us we were going to x equals 2, but we need to figure out this x-coordinate of The function so this y equals x minus x squared What are the text coordinates if we factor out the x we're left with 1 minus x and So we see the x coordinates are 0 which we already noticed and then 1 So we're going to integrate from 0 to 1 In order to proceed we're going to need to foil this thing out All those for all those possible terms first inside first outside inside last So with the typical foil we get 2 pi times integral from 0 to 1 We're going to get a 2x minus 2x squared We're going to get negative x squared and then a positive x cubed dx There are like terms with the quadratics negative 2x and negative x squared a negative 2x squared and then a negative x squared Combining like terms there We end up with a 2x minus 3x squared plus x cubed and so that thing is now ready to integrate Slide this up so we have some more room to write so using the power rule Anti-derivative x square of 2x is an x squared Anti-derivative of the negative 3x squared is a negative x cubed and the anti-derivative x cubed is x to the fourth over 4 Plug in zero and one zero make everything vanish plug in in one powers of one are always one So we've got a bunch a bunch of terms there one of them's a fraction We'll end up with a one minus one plus a fourth The two ones will cancel because one's positive one's negative and so we get 2 pi times a fourth Which is going to give us pi halves as the volume of this solid revolution most excellent there and So much like the washer method we can adapt for alternative Axis of revolution right it doesn't always have to be the x-axis of the y-axis For the washer method you have to be very careful though because you have this inner radius and this outer radius Both of which are affected by the location of the axis revolution In this one since we're only looking the average radius The average radius is just going to be the location of it in this case x equals 2 and then you subtract from it The with the location of the cross section, which is x So you're shifting the axis away from the y-axis doesn't have to it doesn't have to complicate the calculation that much Instead of having an R in this location. Oh, I mean it's still the race instead of an x you get something like 2 minus x and so You'll you'll see some of these things in the homework, of course just make a slight adaptation if your radius Is something other than the y-axis and it doesn't it doesn't it's amazing how it doesn't actually complicate The integral that much just making a slight change It's pretty powerful and so it's quite impressive how versatile the shell method is for this sort of shifting of the axis there