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# Calculating integral disk around vertical line

Video transcript

In the last video, we had
set up the definite integral to evaluate the volume of
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 expression, this
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 to 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-- 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 practiced in this and can do this in your head. You have y plus 1
raised to the 1/2 power. Derivative of y
plus 1 is just 1, which is essentially out here. So if you did u
substitution, you would 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 3/2 multiplied by
the reciprocal 2/3. 2/3 times 4 is 8/3. So it's plus 8/3 times
y plus 1 to the 3/2. And you can verify. If you take the
derivative here you will get this expression
right over here. 3/2 times 8/3 is 4. Decremented, you have y
plus 1 to the 1/2 power. And then finally,
you have-- let's see. What color have I not used yet? Finally, you have this 5. The antiderivative
of 5 is just 5y. And we are going to evaluate
it at 3 and at negative 1, y equals 3 and y
equals negative 1. So this is going
to be equal to pi. So let's evaluate all
this business at 3. So 3 squared over 2 is 9/2. 3 plus 1 is 4 to the 3/2. Well, that's-- so let's see. If square root of 4 is 2 to the
third power is 8, 8 times 8/3 is 64/3, so plus 64/3. 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 3/2 power. That's going to be 0 times 8/3. This 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, that's just
going to be negative 5. And we are in the home stretch. We really just have to do
a little bit of arithmetic, 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. Our least common multiple
of all of these denominators is going to be 6. So let's put everything
over a denominator of 6. So 9/2 is the same
thing as 27/6. 64/3 is the same thing as 128/6. 15 is the same thing as 90/6. 1/2 is the same thing as 3/6. So you would distribute
the negative sign. So this is negative 3/6. And negative times
negative is positive. 5 is the same thing
as 30/6, so plus 30/6. 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 what the numerator is. So let's see if I can
do this in my head. So 27 plus 128 is going
to be-- let me see. That's going to be 140, 155? Is that right? 155? Let's see, if we get to 48
plus another 7-- yeah, 155. Plus 90 gets us to 245. Is that right? Yeah. Plus 90 gets us to 245. Minus 3, you subtract 3
from that, you get to 242. And then you add 30 to
that, you get to 272. So we're left with 272 pi/6. But then we can-- let's see. 272 and 6 are both
divisible by 2. So this is equal to-- let's see. 272 divided by 2 is
going to be 136 pi over-- and if you divide this
denominator right over here by 2-- over 3. Is that right? Yeah. 136 pi/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.