Evaluating integral for shell method example

where we left off in the last video we had set up a definite integral using the shell method for this strange solid of revolution so now let's just evaluate the integral and like we've seen many times with these types of problems we really just have to do some some polynomial multiplication right over here so X -3 squared well that's pretty straightforward that's going to be x squared minus 6x plus 9 and we're going to multiply that times X minus 1 so let's do that first so multiply that times X minus 1 and so negative 1 times 9 is negative 9 negative 1 times negative 6 is positive 6 positive 6 X negative 1 times x squared is negative x squared now x times 9 is 9 x x times negative 6x is negative 6 x squared and then x times x squared is X to the third X to the third and so we get we get X to the third minus 7x squared + 15 X minus 9 and then we have 2 so we just multiply 2 X minus 3 squared times X minus 1 and then we have to multiply that times X so it can essentially raise the degree of each of these things so our original integral has simplified to or at least this be it's easier to now to take the antiderivative it's equal to 2 pi times the definite integral from 1 to 3 of this stuff times X so it's going to be it's going to be X to the fourth X to the fourth minus 7 X to the third power + 15 x squared minus 9 X and then of course DX and I'll make the DX I'll make in that same nice blue color now let's just take the antiderivative so this is going to be equal to 2 pi times the antiderivative of all of this business the antiderivative we're going to evaluate it at 3 and subtract it when it's evaluated at 1 so the antiderivative of X to the fourth is X to the fifth over 5 X to the fifth over five the antiderivative of X to the third is X to the fourth over four and we're going to multiply that times negative seven so it's negative 7x to the fourth over four and then the antiderivative of fifteen x squared that's going to be 15 times X to the third over 3 15 divided by 3 is 5 so it's plus five X to the third and then finally the antiderivative I'm running out of colors of negative 9x that's going to be negative nine halves x squared negative 9x squared over two and you can verify if you take the derivative of this you get this business right over here and so this is going to be equal to this is going to be equal to 2 pi 2 pi and so let's evaluate all of this business let's first evaluate all of this business at 3 so when you evaluate at 3 you have 3 to the fifth power over 5 and I believe that's three to the fifth is 243 but I'll verify 3 to the third is 27 3 to the fourth is equal to 81 3 to the fifth is 243 so this is going to give us some hairy math to deal with it so it's going to be 243 over 5 3 to the fourth power that's 81 but then we have to multiply 81 times 7 81 times 7 so we're going to get 567 that right 81 times 7 7 times 1 is 7 7 times 8 is 56 so we're going to get minus 567 over 4 this is going to be really painful to do the arithmetic part but we'll we'll power through it and then we have 5x to the third X to the third is 27 27 times 5 is 100 and 135 let me just I don't want to make any mistakes 27 times 5 7 times 5 is 35 2 times 5 is 10 yep 135 so plus 135 135 and then finally we have minus 9x squared so x squared is 9 times 9 is 81 so minus 81 over 2 so that's all that's all of this business evaluated at 3 and from that we're going to subtract it when it's evaluated at 1 so we're going to subtract when it's evaluated at 1 and we get let's do this so we get 1/5 1/5 minus 7/4 minus 7/4 minus 7/4 plus 5 plus 5 and then minus 9 halves minus 9 halves minus 9 halves and what are we are left with is just a really hairy fractions problem so I will just hope that I don't make a careless mistake at this point so let's try to do this this is going to be equal to 2 pi times and if we wanted to find a common multiple here it looks like it would have to be 20 least common multiple of 5 and 4 and 2 is 20 so this is going to give us so 2 43 over 5 is the same thing 243 times 4 243 times 4 is going to give us 3 times 4 is 12 4 times 4 is 16 plus 1 is 17 2 times 4 is 8 plus 1 is 2 plus 1 is 9 so we have 9 72 9 72 over 20 and then we have to multiply 567 times 5 so 567 times 5 so you can see the arithmetic is the most painful part here 7 times 5 is 35 6 times 5 is 30 plus 3 is 33 5 times 5 is 25 plus 3 is 28 so we have 28 35 28 35 over four and then 135 over 20 well 135 times 2 is going to be 270 and then times another 10 is 700 so plus 2700 over 20 did I do that right yeah that's right and then finally 81 over 2 is that's going to be the same thing as negative 810 negative let me do that same color negative 810 negative 8 10 over 20 numerator denominator both multiplied by 10 and then let's see negative 1/5 that's the same thing as negative 4 20s and then negative or it's going to be positive 7/4 it's it's the same thing as positive 35 20 it's and there's going to be a negative 5 a negative 5 is the same thing as negative 100 over 20 and then finally the negative nine-halves is the same thing as negative 90 over oh sorry it's going to be a positive I don't make that careless mistake I'll make sure I get the signs right after distribute this negative is going to be a positive 9 halves which is the same thing as plus 90 plus 90 over 20 did I do all the signs right negative 1/5 positive for this one so then positive 7/4 negative 5 and then positive 90 over 20 and so now I just have to do some hardcore addition so let's let's do it so first all I'll take all of the positives and then I'll subtract out the negatives just to simplify it so I have to minimize the number of times well I'll add all the positives together then I'll add all the negatives together and then that'll make it one subtraction problem so if 972 plus 27 hundred plus 35 plus 90 so let me just write it down so this is 2700 plus 972 I should probably take out a calculator at this point but well yeah I'll just do it by hand since I've already done it so much of it by hand Plus 972 plus 35 or plus 90 plus 35 gets us 2 so we get a 7 7 plus 9 is 16 plus 3 is 19 I do that right yeah 16 plus 3 is 19 and this is 17 and this is a 3 so we have 37 97 when we added all the positive numerators and then all the negative numerators let's see I'm going to add them together to see how negative so 28 35 810 4 and 100 so if I add 28 35 810 4 or let me see 100 104 these are all going to be this is how much negative I have to subtract from that 5 plus 4 is 9 3 plus 1 is 4 8 plus 8 plus 1 is 17 and then you have a 3 so we're going to subtract 3797 we're gonna subtract 37 49 from 37 97 and so this actually works out quite well that gets us to 48 so this is equal to 48 let me make sure I haven't made a careless mistake 28 35 810 104 those are all the things I'm subtracting 20 709 70 to 90 and 35 or all the things I added yep 3797 minus 37 49 is going to be equal to 48 so this whole expression we deserve a drum roll now is going to be equal to 2 pi times 48 over over 20 and both 48 and 20 are divisible by 4 so you get 12 12 over 5 my brain is turning into mush now I'm becoming paranoid that I've been making careless mistakes but let's let's we're almost there so it becomes 12 over 5 and so our final answer 12 times 2 is 24 so it becomes 24 PI over 5 and we are done