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# Calculating integral disc around vertical line

AP.CALC:
CHA‑5 (EU)
,
CHA‑5.C (LO)
,
CHA‑5.C.2 (EK)

## Video transcript

in the last video we had set up the definite integral to evaluate the volume of this this upside-down gumdrop truffle looking thing so now in this video we can actually evaluate the definite integral so what we need to do is really just expand out this this expression the square root of y plus 1 plus 2 so let's do that so this is going to be equal to PI times the definite integral from Y is equal to negative 1/2 Y is equal to 3 if you expand this out you get square root of y plus 1 squared which is just going to be Y plus 1 and then you're going to have 2 times the product of both of these terms 2 times 2 times square root of y plus 1 is going to be plus 4 times the square root of y plus 1 and then you have 2 squared so plus 4 so you have this whole thing times dy we can simplify a little bit you have a 1 plus a 4 we can add the 1 to the 4 and get a 5 and now we're ready to take the antiderivative so this is going to be equal to PI times let's take the antiderivative of all of this business PI times pi times and I'll color code it the antiderivative of Y is just Y squared over 2 the antiderivative of 4 times the square root of y plus 1 you just really have to think of it as 4 times y plus 1 to the 1/2 power we could use u-substitution explicitly but you probably are pretty practising this can do this in your head you have y plus 1 raised to the one half power derivative of y plus 1 is just 1 which is essentially out here so you can really just if you did use substitution you would call you'd say u is equal to y plus 1 but this antiderivative is going to be equal to well if you increment this exponent you get 3 halves x the reciprocal 2/3 2/3 times 4 is 8 thirds so it's plus 8/3 times y plus 1 to the three-halves and you can verify if you take the derivative here you will get this expression right over here three-halves times 8 thirds is 4 decremented you have y plus 1 to the 1/2 power and then finally you have finally you have let's see what color if I not used yet finally you have this 5 the antiderivative of 5 is just 5 y 5 y and we are evaluating it we are going to evaluate it from at three and at negative one y is equals 3 and y equals negative one so this is going to be equal to PI so let's evaluate all this business at 3 so 3 squared over 2 is 9 halves 3 plus 1 is 4 to the three-halves well that's let's see if square root of 4 is 2 to the third power is 8 8 times 8 thirds is 64 over 3 so plus 64 over 3 you have 5 times 3 well that's going to be 15 plus 15 and from that we're going to subtract all this business evaluated at negative 1 so you have negative 1 squared over 2 well that's just 1/2 negative 1 plus 1 is 0 to the 3 1/2 power that's going to be 0 times 8 thirds is all going to be 0 so we don't have to even write it and then finally you have negative 1 times 5 well this is going to be negative 5 negative 5 and we are in the homestretch we really just have to do a little bit of arithmetic add some add some hairy fractions right over here so let's do it so this whole thing is going to simplify to PI times and it looks like let's see or at least common multiple of all of these denominators is going to be 6 so let's put everything over a denominator of 6 so 9 halves is the same thing as 27 over 660 4/3 is the same thing as 128 over 6 15 15 is the same thing as 90 over 6 90 over 6 1/2 is the same thing as 3 over 6 so this is distribute the negative signs this is negative 3 6 and you just and negative times negative is positive 5 is the same thing as 30 over 6 so plus plus 30 over 6 and so this is going to give us our denominator is going to be over 6 we're going to multiply something times pi we have this PI over here and then we just have to figure out the numerator is so let's see if I can do this in let's see if I can do this in my head so 27 plus 1 28 27 Plus 128 is going to be let me see that's going to be 140 155 is that right 155 is let's see if we get to 48 plus another seven yet 155 plus 90 gets us to 245 is that right yeah plus 96 to 245 minus 3 you subtract 3 from that you get to 242 and then you add 32 that you get to - 72 - 72 so we are left with 272 PI over 6 but then we can let's see - 72 + 6 are both divisible by 2 so this is equal to C - 72 divided by 2 is going to be 1 1 36 PI over and if you divide this denominator right over here by 2 over 3 is that right yeah 136 PI over 3 and 136 is not divisible by 3 so we have it as simplified as we can this right over here is the volume of our little upside-down gumdrop looking thing
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