More volume questions, integration

• Feb 28th 2013, 12:41 PM
togo
More volume questions, integration
Question 26-3-15
Statement
Using Shell method, find the volume generated by revolving the region bounded by the given curve about the x-axis.
x = 4y - y^2 - 3, x = 0

Finding y intecept
0 = 4y - y^2 - 3
Y intercept: 1

Shell method: 2pixy
= 2pi(4y-y^2-3)y
= 2pi(4y^2-y^3-3y)

When I calculate this now with y = 1 the answer is zero, which is wrong.

Question 26-3-21
Statement
Using shell method, find the volume generated by revolving the region bounded by the given curve about the y axis.
x^2 - 4y^2 = 4, x = 3

Attempt
2pixy principal formula
Isolate y
-4y^2 = 4 - x^2
-y^2 = (4-x^2)/4

-y^2 = 1 - (x^2/4)
y = 1 - x/2

Now to integrate
= 2pix(1 - x/2)
= 2pix - (x^2/2)

not sure how to proceed.

Thanks for looking
• Feb 28th 2013, 01:41 PM
togo
Re: More volume questions, integration
I will add a few attempts:

26-3-8 Rotate about x-axis, use shell method, determine volume:
y = x^(1/2), x = 0, y = 2
dV = 2pixy
Isolate x
x = y^2
dV = 2pi(y^2)y
dV = 2piy^3
dV = 1/4y^4
dV = 2pi(1/4(2^4)) = 25.132

Is this correct? Thanks.

26-3-10 Rotate about x-axis, use disk method, determine volume:
y = 4x - x^2
y = 0
dV = piy^2
y = (4x-x^2)(4x-x^2)
y = 16x^2 - 4x^3 - 4x^3 + x^4
y = 16x^2 - 8x^3 + x^4

0 = 4x-x^2

I'm stuck trying to find a limit here.

26-3-12 Rotate about x-axis, use shell method, determine volume:
y = x^2, y = x
dV = 2piy^2
y^2 = (x^2)^2
y^2 = x^4
1/4x^4

is this the correct path to take? Thanks.
• Feb 28th 2013, 06:12 PM
Prove It
Re: More volume questions, integration
1. To rotate a region about the x axis, use \displaystyle \begin{align*} V = \int_a^b{\pi \left[ f(x) \right] ^2 \, dx} \end{align*}

By a simple sketch, it's relatively easy to see that you will need to evaluate a large volume and subtract a smaller one. So your volume is evaluated by

\displaystyle \begin{align*} V = \int_0^1{ \pi \left( 2 + \sqrt{1 - x} \right)^2 \, dx } - \int_0^1{ \pi \left( 2 - \sqrt{1 - x} \right)^2 \, dx} \end{align*}.

Use this formula for the other problems too.
• Feb 28th 2013, 10:37 PM
togo
Re: More volume questions, integration
uh

ok, hate to say it but I'm still pretty lost, probably find a tutor next... i spent 2 hrs on this stuff this morning and its getting harder and harder. I got thru the first few questions easily enough though.

or maybe I am really tired and need to look at it with fresh brain. I kinda see where you're going by subtracting volumes.
• Mar 3rd 2013, 04:54 PM
ThomasHornbeck
Re: More volume questions, integration
Togo,

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