in this video we're going to focus on finding the volume using the shell method so let's begin by drawing a picture now we're going to find the volume when rotated about the y-axis what i'd like to do is draw a rectangle the limits of integration is going to be from a to b the radius is the distance between the x-axis and the axis of rotation by the way you should always draw the rectangle parallel to the axis of rotation so in this case you want it to be parallel to the y axis and then this is going to be the height of the shell or the rectangle so using the shell method the equation is this the volume is going to be equal to 2 pi integration from a to b and then it's the radius which is going to be in terms of x times the height in terms of x so if you want to find the volume of the solid when rotated about the y axis the radius and the height have to be in terms of x now sometimes you may have two curves as opposed to one let's call the top curve f of x and the bottom g of x so we want to rotate it about the y axis so the rectangle is going to be parallel to the y axis now the radius is still the distance between the rectangle and the axis of rotation and this is going to be the height of the shell the only difference in this situation is that the height h of x is basically equal to the top function f of x minus the bottom function g of x everything else being the same the limits of integration will still be a to b they will represent x values now sometimes you may need to find a volume when rotated about the x-axis so let's use the same type of picture but we're going to rotate the region about the x-axis this time instead of the y-axis so now we're going to draw the rectangle parallel to the axis of rotation that is parallel to the x-axis the radius is the distance between the rectangle and the axis of rotation the height is parallel to the axis of rotation so this is the height of the rectangle so the volume is going to be pi well 2 pi not just pi and this time the limits of integration is c to d it's going to be y values so 2 pi integration from c to d of the radius in terms of y times the height in terms of y d1 so keep this in mind if you want to rotate the curve about the x-axis the radius and the height has to be in terms of x for this to work now let's see if we have two functions f and g let's say the function on the right is f and the one on the left is g we're going to rotate about the x axis so let's draw the rectangle parallel to the x axis this is going to be the radius still and here we have the height so the height is going to be equal to the function on the right which is f of x minus the function on the left g of x and that's it that's how you can find h so let's start with the first example let's say that we have the curve y is equal to the square root of x and a region is bounded by that curve the line y equals zero and x equals four and we're going to rotate it about the y axis so feel free to pause the video and try this problem so let's begin by drawing a picture so this is the square root of x and it's going to stop at x equals four so let's draw a rectangle and we're going to rotate this about the y axis so this is the radius and this is the height so notice that the radius is the same as x so thus we can say that r is equal to x x is the distance between the y axis and the rectangle notice that the height h is the same as y so we can say that h is equal to y now if we're rotating about the y axis or about any line parallel to the y axis the radius and the height has to be in terms of x now the radius is already in terms of x which is good we just got to convert y into something in terms of x now we know that y is equal to the square root of x so we can replace y with root x so now we have h in terms of x now let's use the formula so the volume is equal to 2 pi integration from a to b r of x times h of x dx a is zero b is four so we're going to integrate it from zero to four the radius is x the height is the square root of x x is basically x to the first power square root x is x to the one half whenever you multiply by common bases you need to add the exponents so one plus one half that's basically two over two plus one over two that's uh three over two so therefore we can rewrite the expression like this this is what we have so far so now let's integrate it the anti-derivative of x to the three half we need to add one to the exponent three over two plus one is five over two and instead of dividing by five over two let's multiply by two over five let's evaluate it from zero to four so first let's plug in four so we have four raised to the five over two multiplied by two over five and then minus two times zero raised to the five over two over five which that whole thing's gonna be zero now what is four raised to the five over two this is the same as four to the half raised to the fifth power whenever you raise one exponent to another you need to multiply the two exponents one half times five is five over two now four to the one half power is the same as the square root of four so that's two and two to the fifth power is 32 so we have two pi 2 times 32 over 5 minus 0 and 2 times 32 is 64 and 2 times 64 is 128. so the final answer is 128 pi divided by five and so uh that's the solution now let's work on another example let's say that y is equal to x minus x cubed and the region is bounded by that curve the line y equals zero and it's from x equals zero to x equal one and we're going to rotate this region about the y axis so try this example see if you can get the right answer using what you know already now what we can do is let's find the x-intercepts for this graph let's set y equal to zero if we factor out an x we're gonna have one minus x squared left over and we could factor one minus x squared using the difference of perfect squares method it's one plus x and one minus x the x-intercepts are zero negative one and one it turns out that if you graph this function and focus in only on the right side of the x i mean the right side of the y axis this graph is going to look something like this where this is zero and this is one and we're going to rotate it about the y axis and so let's draw a rectangle that's parallel to the y axis so this is going to be the radius and here we have the height so just like before the radius is the same as x and the height we can see is y so r equals x and h is equal to y but we need the radius and the height to be in terms of x so let's replace y with x minus x cubed so that's going to be the height so now we can write the integral the volume is going to be 2 pi integration from a to b or 0 to 1 and then the radius times the height dx so what we're going to do at this point is distribute x x times x is x squared x times x cube is x to the fourth so now let's integrate the function the antiderivative of x squared is x cubed over three and the antiderivative of x to the fourth is x to the fifth over five integrated from zero to one and let's not forget to multiply our answer by two pi now let's begin by plugging in one one to the third is just one so it's going to be one over three and one to the fifth is one so one over five and then if we plug in zero zero cube is zero zero to the fifth is zero so the whole thing is zero now we need to get common denominators so let's multiply one over three by five over five and then let's multiply one over five by three over three so this will give us a common denominator of fifteen so what we have now is five over fifteen minus three over fifteen five minus three is two so this is two over fifteen times 2 pi 2 times 2 pi is 4 pi so the final answer to this problem it's 4 pi over 15.