Finding the volume of a solid. I need help.

[sweetwater88]

My final exam is tomorrow and my teach is "away" on her vacation. I need to learn how to do finding the volume of a solid though. At least some help....

So: Can way one help me step by step? Please?

For example: Finding the volume of the solid formed by rotating the region y = x^2, and y = 4 about the line y = -1.

Please, any help is good help!

 

[galactus]

Since you're rotating about y=-1, you might try shells.

Did you graph your functions?.

\(\displaystyle \L\\{4\pi}\int_{0}^{4}(y+1)\sqrt{y}dy\)

Maybe you can try it with slices and see if you get the same thing.

 

[soroban]

Hello, sweetwater88!

Find the volume of the solid formed by rotating the region \(\displaystyle y\,=\,x^2,\;y\,=\,4\) about the line \(\displaystyle y\,=\,-1\)

          *       |       *
                  |
        - -*- - -4+ - - -* - -
            *:::::|:::::*
              *:::|:::*
        ---------***---------
                  |
      - - - - - -1+ - - - - - -
                  |

We will use "washers": \(\displaystyle \;V\;=\;\pi\L\int\)\(\displaystyle [(\text{outer radius})^2\,-\,(\text{inner radius})^2]\,dx\)

The outer radius is: \(\displaystyle \,4\,+\,1\:=\:5\)

The inner radius is: \(\displaystyle y\,+\,1\:=\:x^2\,+\,1\)


And we have: \(\displaystyle \L V \;= \; \pi\int^{\;\;\;2}_{-2}\left[5^2\,-\,(x^2\,+\,1)^2\right]\,dx\)