Hello, calc101!

Did you make a sketch?

Your radii are wrong . . .

Consider the solid obtained by rotating the region bounded by the given curves

about the line $\displaystyle x = 5.$ .The curves are: $\displaystyle y\:=\:\sqrt{x}\text{ and }y\:=\:x$.

Find the volume $\displaystyle V$ of this solid.

Facts: The only points of interesction are at (0,0) and (1,1).

Attempted Solution:

We need to use the washer method.

However, since the region is being rotated around the line $\displaystyle x=5$,

the radius of each circle will be $\displaystyle r = r + 5$. . . . . no

We also need to integrate from 0 to 1 along y. Code:

| :
| *
| ...* :
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|*::/ :
|:/ :
- * - - - - - - - - + - -
| 5

Our functions are: .$\displaystyle x \,=\,y^2,\;x \,=\,y$

The outer radius is: .$\displaystyle r_1 \:=\:5-y^2$

The inner radius is: .$\displaystyle r_2 \:=\:5-y$

Hence: .$\displaystyle V \;=\;\pi\int^1_0(5-y^2)^2\,dy - \pi\int^1_0(5-y)^2\,dy $