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# Evaluating the integral for washer method

Video transcript

Using the-- I guess we
could call it the washer method or the ring
method, we were able to come up with
the definite integral for the volume of this solid
of revolution right over here. So this is equal to the volume. And so in this
video, let's actually evaluate this integral. So the first thing that we could
do is maybe factor out this pi. So this is going to be equal to
pi times the definite integral from 0 to 1. And then let's square this
stuff that we have right here in green. So 2 squared is going to be 4. And then we're going to have
2 times the product of both of these terms. So 2 times negative
y squared times 2 is going to be
negative 4 y squared. And then negative
y squared squared is plus y to the fourth. And then from that, we are going
to subtract this thing squared. We're going to subtract
this business squared, which is going to be 4 minus
4 square roots of y plus-- well, square root of y
squared is just going to be y. And all of that dy. Let me write that
in that same color. And so this is going
to be equal to pi times the definite
integral from 0 to 1. And let's see if we
can simplify this. We have a positive
4 here, but then when you distribute
this negative, you're going to have a negative
4, so that cancels with that. And let's see. The highest-degree
term here is going to be our y to the fourth. So we have a y to the fourth. I'll write it in
that same color. And so the
next-highest-degree term right here is this
negative 4 y squared. So then you have negative--
let me do that same color. We have negative 4 y squared. That's that one
right over there. And then we have this y. But we have to remember we
have this negative out front. So it's a negative y. So this one right over
here is a negative y. And then we have a
negative times a negative, which is going to give us a
positive 4 square roots of y. So this is going to end up being
a positive 4 square roots of y. And actually, just
to make it clear when we take the antiderivative,
I'm going to write that as 4 y to the 1/2 power. And we're going to multiply
all that stuff by dy. Now we're ready to take
the antiderivative. So it's going to be equal to pi
times the antiderivative of y to the fourth is y
to the fifth over 5. Antiderivative of negative 4
y squared is negative 4/3 y to the third power. Antiderivative of negative y
is negative y squared over 2. And then the antiderivative
of 4 y to the 1/2-- let's see. We're going to
increment, so it's going to be y to the
3/2 multiplied by 2/3. We're going to get 8/3
plus 8/3 y to the 3/2. And let's see. Yep. That all works out. And we're going to evaluate
this at 1 and at 0. And lucky for us, when
you evaluate at 0, this whole thing
turns out to be 0. So this is all going to be
equal to pi times evaluating all this business at 1. So that's going to
be 1/5 minus 4/3-- I'll do that in a green
color-- minus 4/3 minus 1/2-- so whenever you
evaluate it at 1, it's just going to
be-- so plus 8/3. And let's see. What's the least common
multiple over here? Let's see, a 5, a 3, and a 2. It looks like we're going
to have a denominator of 30. So we can rewrite
this as equal to pi, and we can put everything
over a denominator of 30. 1/5 is 6/30. 4/3 is 40 over 30,
so this is minus 40. That's the different
shade of green. Well, actually, let me make
it another shade of green. So this is minus 40/30. Negative 1/2,
that's minus 15/30. And then finally, 8/3 is
the same thing as 80/30, so that's plus 80. So this simplifies
to-- so let's see. We have 86 minus
50-- oh, actually, let me make sure I'm doing
the math right over here. So 80 minus 40 is going to get
us 40, plus 6 is 46, minus 15 is 31. So this is equal to 31 pi 30. I have a suspicion that I
might have done something shady in this last part
right over here. So this is going
to be, let's see, negative 36,
negative 51, plus 80. I think that seems right. I'm going to do
it one more time. Let's see. 80 minus 40 is 40, 46, 46
minus 10 is 36, minus another 5 is 31. So, yes, we get 31 pi
over 30 for our volume.