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Main content
Current time:0:00Total duration:8:14
AP.CALC:
CHA‑5 (EU)
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CHA‑5.C (LO)
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CHA‑5.C.4 (EK)

Video transcript

where we left off in the last video we'd actually set up our definite integral to figure out the volume of this figure so now we just have to evaluate it and really the hardest part is going to be simplifying that and that right over there so let's get to it so what is this thing squared well it's gonna have to do a little bit of polynomial multiplication here so I'm going to go into that same color I had so we're going to have 4 minus x squared plus 2x times 4 minus actually let me write that in the degree in the order of the the terms or the degree of the term so it's negative x squared plus 2x plus 4 I just switched the order of these things times negative x squared plus 2x plus 4 so we're just going to multiply multiply these two things I shouldn't even write a multiplication symbol looks too much like an X so 4 times 4 is 16 4 times 2x is 8x 4 times negative x squared is negative 4 x squared negative 4x squared 2x times 4 is 8x 2x times 2x is 4x squared then 2 x times negative x squared is negative 2x to the third power and now we just have to multiply negative x times all of this so negative or negative x squared negative x squared times 4 is negative 4x squared negative x squared times 2x is negative 2x to the third power and the negative x squared times negative x squared is positive x to the fourth so it's going to be positive x to the fourth and now we just have to add up all of these terms and we get see we get X to the fourth add these two minus 4x to the third and then this cancels with this but we still have that so minus 4x squared you add these 2 right over here you get plus 16x and then you have plus 16 so that's this expanded out and now if we want to X minus 4 or 4 minus X times 4 minus X so if we just have 4 minus x times 4 minus X we could actually do it this way as well that's just going to be 4 times 4 2:16 time plus 4 times negative X which is negative 4x negative x times 4 another negative 4x and then negative x times negative X is equal to plus x squared so if we were to swap the order we get x squared minus 8x minus 8x plus 16 but we're going to subtract this so if we put that if you have the negative sign out there we're going to subtract this business so let's just let's just do it right over here since we already have it set up so we have this minus this so we have this minus we're going to subtract we're going to subtract this or we could add the negative of it so we'll put negative x squared plus 8x plus 8x minus 16 so I'm just going to add the negative of this and we get we get and I'll do this in a new color we get let's see we get X to the fourth minus 4x to the third power minus let's see minus 5x squared minus 5x squared plus 24x and then these cancel out so that's what we are left with and so that's the inside of our integral so we're going to take the integral of this thing just so I don't have to keep rewriting this thing from zero until if I remember correctly 3 yep from 0 to 3 from 0 to 3 of this DX and then we had a PI out front here we had a PI out front here so I'll just take that out of the integral so times pi times pi well now we just have to take the antiderivative and this is going to be equal to pi times pi times antiderivative of X to the fourth is X to the fifth over five antiderivative 4x to the third is actually X to the fourth so this is going to be minus X to the fourth you can verify that derivative is 4x and then you decrement the exponent 4x to the third so that works out and then the antiderivative of this is negative 5/3 X to the third just incremented the exponent and divided by that and then you have plus let's see plus 24x squared over 2 or 12x squared and we're going to evaluate that I like to match colors for my opening and closing closing parentheses we're going to evaluate that actually let me do it as brackets we're going to evaluate that from 0 to 3 0 to 3 so this is going to be equal to PI times let's evaluate all this business at let's evaluate all this business at 3 so we're going to get 3 to the fifth power 3 to the fifth power so let's see 3 to the 3 to the 3rd 3 to the third is equal to 27 3 to the fourth is equal to 81 3 to the fifth is going to be equal to this times 3 is 243 so it's going to be 243 it's going to be some hairy math 243 over 5 minus well 3 to the 4th is 81 minus 81 minus let's see we have we're going to have 3 to the 3rd tied let's see so it's going to be minus 5 over 3 times 3 to the third power so times 27 well 27 divided by 3 is just 9 9 times 5 is 45 so let's just simplify that so this is going to be equal to 45 did I do that right yeah it's essentially going to be like 3 squared times 5 because you're dividing by 3 here so it's going to have 45 and then finally 3 squared is 9 9 times 12 is 108 so plus 108 these problems that I involve hairy arithmetic are always the most stressful for me but I'll try not to get too too stressed and then we're going to subtract out this whole thing evaluated at 0 but lucky for us that's pretty simple this evaluated 0 0 0 0 0 so we're going to subtract out 0 which simplifies things a good bit so now we just have to do some hairy fraction arithmetic so let's do it so what I'm going to do first is simplify all of this part and then I'm going to worry about putting it over a denominator of 5 so let's see what we have we have negative 81 minus 45 so these 2 right over here become negative one negative 126 and the negative 126 plus 108 well that's just going to be the same thing as negative 26 plus 8 which is going to be negative 18 so this whole thing simplifies to negative 18 did I do that right did I do that right so this is negative 126 and then negative 126 yes it will be negative 18 so now we just have to write negative 18 over 5 negative 18 over 5 is the same thing as negative this is to see 5 times 1 5 times 10 is 50 plus 40 so that's going to be negative 90 over 5 so this whole thing has simplified to it's equal to PI times 2 43 over 5 243 over 5 minus 90 over 5 minus 90 over 5 which is equal to PI guys I deserve a drum roll now this was some hairy mathematics so 243 minus and 90 is going to be 153 153 over 5 or we can write this as 153 PI over 5 and after all that math you sometimes forget what we were doing in the first place we were figuring out this right over here is the volume is the volume of this figure that had this little inside of it bored out
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