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

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.

Re: More volume questions, integration

1. To rotate a region about the x axis, use $\displaystyle \displaystyle \begin{align*} V = \int_a^b{\pi \left[ f(x) \right] ^2 \, dx} \end{align*}$

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

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

Use this formula for the other problems too.

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.

Re: More volume questions, integration

Togo,

I'm the Co-Founder of Tutor Universe

It's a relatively new service but we have some very skilled online math tutors that can help you. I would encourage you to check it out. Let me know if you need anything. You can contact me via email at thomas [ at ] tutoruniverse [ dot ] com.

I would highly recommend Mike Melko, he's a retired Math Professor and has been helping a lot of students. You can view his profile here:

Mike Melko - Tutor Universe

Hope this helps!