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Current time:0:00Total duration:5:11

in the last video we set up this definite integral to evaluate the volume of the solid of revolution that we set up using the shell method so now let's just evaluate this thing so the main thing is just simplifying this expression I'll start off by trying to simplify this part of it so that's going to be y plus 1 I just need an Apple so something we were just happened in my throat but anyway that's done with y plus 1 minus y squared minus 2y plus 1 I just squared expanded out this binomial and then that would simplify to another Apple in my throat moment so that's going to be y plus 1 minus y squared plus 2 y minus 1 so this one and this negative 1 cancel out and let's see you get negative y squared plus 3y plus 3y and then we're going to multiply that we're going to multiply that times times y plus 2 so when you multiply y plus 2 times this so you have Y times negative Y squared gets us negative Y to the third power y times 3y is going to be plus 3y squared 2 times negative Y squared is negative 2y squared and then 2 times 3 y is plus 6y so then you go all the way down here this thing can simplify to because you have 3y squared minus 2 y squared so this is going to be negative Y to the third Plus this part right over here simplifies to just Y squared plus y squared plus 6y so that's this entire part simplified simplifies to this town here we can take the 2 pi out of the integral sign so let's do that we're integrating from Y is equal to 0 to Y is equal to 3 dy and I took the 2 pi out here and that is equal to our volume and so now we're essentially ready to take the antiderivative this is going to be equal to 2 pi times the antiderivative of this business evaluated at 3 minus evaluated at 0 so the antiderivative of and I'll color code it this is I found this useful this is the antiderivative of y 2/3 is Y to the 4th over 4 so this is negative y to the 4th over 4 antiderivative of Y squared is y to the third over 2 or Y to the third over 3 I should say Y to the third over 3 and then finally finally I'll do it in yellow antiderivative of 6 y is 3y squared so plus 3y squared and we are going to evaluate all of this business at 2 0 and 3 so this simplifies this is going to be equal to 2 pi times well let's see let me do the same colors 3 to the fourth power is 81 so it's negative 81 over 4 negative 81 over 4 plus let's see 3 2/3 27 divided by 3 is 9 plus 9 and then 3 squared is 9 times 3 is 27 plus 27 and then when you evaluate all of these things at 0 you just get 0 so you're just subtracting out 0 so we really don't have to do anything else with the 0 and now we are ready to simplify so this turns out to be let me see this is going to turn out to be and actually actually let's just add them all up so this is going to be 9 plus 27 is 36 so that is 36 and if we want to add it to negative 81 over 4 we have to find a common denominator so all of this business is going to be equal to 2 pi times and so our common denominator can be 4 times something over 4 we have negative 81 over 4 and then 36 times 4 is 144 is that right yep that's 144 so 36 times 4 is so it's plus 144 30 times 4 is 120 plus another 24 is 144 so you have 144 essentially minus 81 so this is going to be equal to 2 pi times that actually I can even simplify it a little bit more we have a 2 here and a 4 there so divide the numerator and the denominator by 2 so you get over 2 so you're going to have pi times let's see this is going to be 44 let's see if this was an 80 this would be 64 right that would be 64 so it's going to be 63 so it's going to be 63 let me write this way it's going to be 63 pi 63 PI over 2 did I do that right 60 plus 81 is 141 add another 3 to 144 yep and we're done we figured out the volume of our front of jet engine looking shape