Help! Find the Volume of the solid!

[carlycakes]

Find the Volume of the solid generated by revolving the region about the y-axis.

the region enclosed by x = y^2 / 5 , x = 0, y = -5, y = 5

can you please show me step-by-step process to get the answer! thanks!

 

[mmm4444bot]

Please tell me what you've learned so far about the Disc Method for finding volumes of solids of revolution.

It would also help, if you could say something about what you've already done OR where you're stuck.

Thanks

PS: This appears to be two identical objects, not one, since there is no volume at the Origin. We can integrate from y = 0 to y = 5, and double the result.

 

[carlycakes]

im stuck on the problem...the disk method is soo confusing.
i don't know where to start.

 

[soroban]

Hello, carlycakes!

Find the Volume of the solid generated by revolving the region about the y-axis.

. . \(\displaystyle x = \frac{y^2}{5},\;x = 0,\; y = -5,\; y = 5\)

First, make a sketch . . .

       5|
        |:::::::::*
        |:::::*
        |::*
        |*
    - - * - - - - - -
        |*
        |::*
        |:::::*
        |:::::::::*
      -5|

How can "discs" be confusing?
It's the simplest of all the Volume-of-Revolution problems . . .

\(\displaystyle \text{Formula: }\;V \;=\;\pi\!\! \int^b_a\!\! x^2\,dy\)


\(\displaystyle \text{We have: }\;V \;\;=\;\;\pi\!\int^5_{\text{-}5}\!\left(\frac{y^2}{5}\right)^2dy \;\;=\;\;\frac{\pi}{25}\!\int^5_{\text{-}5}\!y^4\,dy \;\;=\;\; \frac{\pi}{125}y^5\,\bigg]^5_{\text{-}5}\)

. . . . . . . . \(\displaystyle =\;\;\frac{\pi}{125}\bigg[(5^5) - (-5)^5\bigg] \;\;=\;\; \frac{\pi}{5^3}\left(2\cdot5^5\right) \;\;=\;\;\pi\cdot2\cdot5^2 \;\;=\;\;50\pi\)

 

[BigGlenntheHeavy]

\(\displaystyle Given: \ x \ = \ \frac{y^2}{5}, \ y \ = \ \pm5, \ x \ = \ 0, \ find \ volume \ by \ rotating \ along \ the \ y-axis.\)

\(\displaystyle Method \ 1, \ disks: \ 2\pi\int_{0}^{5}\bigg(\frac{y^2}{5}\bigg)^2dy \ = \ 50\pi.\)

\(\displaystyle Method \ 2, \ shell: \ 4\pi\int_{0}^{5}x(5-\sqrt{5x})dx \ = \ 50\pi.\)

\(\displaystyle See \ graph.\)

[attachment=0:2e2gfc6y]eee.jpg[/attachment:2e2gfc6y]

 

[carlycakes]

thanks. i think i understand now