Disk Method about the x axis

[crzyberzu123]

Does anyone know how to solve x(5-x)^(1/2) about the x axis using the disk method?

 

[soroban]

Hello, crzyberzu123!

\(\displaystyle \text{Does anyone know how to solve }y \:=\:x(5-x)^{\frac{1}{2}}\text{ about the }x\text{-axis using the disk method?}\)

I must assume that the specified region is in Quadrant 1.

The graph looks like this:

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

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

\(\displaystyle \displaystyle\text{So you have: }\:V \;=\;\pi\!\int^5_0\!\!\left[x\sqrt{5-x}\right]^2\,dx\)

Got it?

 

[crzyberzu123]

Hello, crzyberzu123!


I must assume that the specified region is in Quadrant 1.

The graph looks like this:

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

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

\(\displaystyle \displaystyle\text{So you have: }\:V \;=\;\pi\!\int^5_0\!\!\left[x\sqrt{5-x}\right]^2\,dx\)

Got it?

Hello!
thank you for your response.
However, would you make the equation equal to x first?

 

[srmichael]

Hello!
thank you for your response.
However, would you make the equation equal to x first?

No, not in this case since you are specifically being asked to do this using the disk method. Thus, the respective rectangles that you are summing up from 0 to 5 are perpendicular to the x-axis and thus you would use dx in your integral and thus the function in terms of x.

If you were asked to solve this same volume, but instead using cylindrical shells, then you would have to solve the equation in terms of y.