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### Course: AP®︎/College Calculus AB > Unit 8

Lesson 12: Volume with washer method: revolving around other axes- Washer method rotating around horizontal line (not x-axis), part 1
- Washer method rotating around horizontal line (not x-axis), part 2
- Washer method rotating around vertical line (not y-axis), part 1
- Washer method rotating around vertical line (not y-axis), part 2
- Washer method: revolving around other axes

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# Washer method rotating around vertical line (not y-axis), part 2

Evaluating the integral set up in the last video using washer method. Created by Sal Khan.

## Want to join the conversation?

- Is there a chance that the final answer may some how turn out to be negative?

Does this mean this answer is wrong? Or do we need to take the absolute value in these cases?(5 votes)- For these particular integrals, it isn't possible for the final answer to be negative. This is then an easy check that the answer is wrong if you see a negative number as the final result.(10 votes)

- On the AP Exam, in the free response section, could you just leave the answer as "pi((1/5)-(4/3)-(1/2)+(8/3)" or do you need to simplify further?(2 votes)
- How you left it would get you full credit! A good rule of thumb is that if you can type it into a basic calculator and get a number, you don't have to simplify it further.(7 votes)

- shouldn't integration between 0 to 4 because hell shape covers that part(1 vote)
- No. The
`2D`

slice in the rotation is where the range of the integral will be defined, and not the rotation envelope.(6 votes)

- What do you do if you are rotating around a line such as x=1 yet you have area bound in the negative x direction? Meaning if you rotate it around the line x=one you may have 2 points where x equals 1.5 due to a parabola shape or something similar(2 votes)
- My answer was completely different to Khans, during my evaluating part I did:

(2-y^2)^2 = 4 - 2y^2 - 2y^2 + y^4

Simplifying down to= -4y^4 + y^4 +4 (this is where Khan stopped simplifying, I took it further)

Down to= -3y^4 + 4

The second binomial was the same as khan: (2-2(Squrt(y)) -----> -4Squrt(y) + y + 4

Integrating:

(Pi) intervals from 1-0 ((-3y^4 + 4) - (-4SquaRt(y) + y + 4))dy

= (Pi) intervals from 1-0 (-3y^5/5 + 8/3y^3/2 - y^2/2) dy

Summing boundaries:

= Pi((-3(1)^5/5 + 8/3(1)^3/2 - (1^2)/2) - 0))

=Pi(-3/5 + 8/3 - 1/2) (We will use 30 as our common denominator)

There fore = Pi((-18+80-15)/30))

= 47Pi/30, wahh la!!(0 votes)- You simplified incorrectly: - 2y² - 2y² = -4y² rather than -4y⁴.... you had the wrong exponent. Thus, everything you did from that point on is not correct.(5 votes)

- why does he use 1 as his top bound?(2 votes)
- I cant say specifically because I don't know the equation, however it means that it either intersects the line x=1 or that is where it crosses the x-axis(1 vote)

- is it always calculated from right to left?(1 vote)
- Sometimes I am confused with the reign I need to rotate.Just like this one:What's the volume if I rotate the area between x=1,y=4,and y=x^2 in the first quadrant.I took the integral between x=0 and x=1.And I got it wrong.How can I find the reign correctly?(1 vote)
- Please specify what line or axis you are rotating about. If you are rotating about y=4, the bounds of integration are x = 1 and x = 2. If you are rotating about x = 1, the bounds of integration are y = 1 and y = 4. Hope that I helped.(1 vote)

- Can we use the washer method rotating around a function,not just any vertical line or horizontal line?(1 vote)
- Tl;dr: It's complicated.

See previous answers in the comments of other videos. Here is a great link: http://www.sfasu.edu/honors/urc/docs/2012/Moyer_and_Robinson.pdf(1 vote)

- omg finally done with this! i can now FINALLY move on to multivariable.(1 vote)

## 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.