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Studying for a test? Prepare with these 8 lessons on Applications of definite integrals.

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# Part 2 of shell method with 2 functions of y

Video transcript

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 ate an apple,
so something weird just happened in my throat. But anyway, that's done with. y plus 1 minus y
squared minus 2y plus 1. I just 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 2y minus 1. So this 1 and this
negative 1 cancel out. And let's see. You get negative
y squared plus 3y. And then we're going to
multiply that times y plus 2. So when you multiply
y plus 2 times this, so you have y times
negative y squared, it 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 3y is plus 6y. So then you go all
the way down here. This thing can simplify,
too, because you have 3y squared minus 2y squared. So this going to be
negative y to the third plus-- this part right
over here simplifies to just y squared--
y squared plus 6y. So that's this entire part
simplified to this down 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. And I'll color code it. I found this useful. The antiderivative
of y to the third is y to the fourth
over 4, so this is negative y to
the fourth over 4. Antiderivative of y squared is
y to the third over 2-- or y to the third over
3, I should say. And then finally,
I'll do it in yellow. Antiderivative of 6y is 3y
squared, so plus 3y squared. And we are going to evaluate
all of this business at 0 and 3. So this simplifies. This going to be equal to 2
pi times-- well, let's see. Let me do it in the same colors. 3 to the fourth power is 81. So it's negative
81 over 4, plus-- 3 to the third is 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. Let me see. 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 just 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? Yeah, that's 144. So 36 times 4, 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-- and actually, I can
even simplify it little bit more, because we
have a 2 here and a 4 there. So divide the numerator
and denominator by 2 so you get it over 2. So you're going to have pi
times-- this is going to be 44. Let's see. If this was an 80,
this would be 64. So it's going to be 63. Let me write it this way. It's going to be 63 pi over 2. Did I do that right? 60 plus 81 is 141. Add another 3, you get a 144. Yep. And we're done. We figured out the
volume of our front of jet engine-looking shape.