Originally Posted by

**angel.white** Question: Let V be the volume of the solid obtained by rotating about the y-axis the region bounded by $\displaystyle y=\sqrt{x}$ and $\displaystyle y=x^2$. Find V both by slicing and by cylindrical shells. In both cases, draw a diagram to explain your method.

My work:

-------------------

**a) Volume by slices.**

Since this is rotated about the y-axis, I solved in terms of x:

$\displaystyle y=\sqrt{x} ~~~~\Rightarrow ~~~~x=y^2$

$\displaystyle y=x^2 ~~~~~\Rightarrow ~~~~x = \sqrt{y}$

Then I said the cross-sectional areal will be $\displaystyle \pi x^2$ since x will be the radius.

So I plugged my equations into it:

$\displaystyle Area_1 = \pi y$

$\displaystyle Area_2 = \pi y^2$

Since they intersect at the points (0,0) and (1,1) Volume will be from y=0 to y=1:

$\displaystyle V=\int_0^1 (Area_1-Area_2)~dy$

$\displaystyle V=\pi \int_0^1 (y-y^2)~dy$

$\displaystyle V=\pi (\frac 12y^2-\frac 13y^3)_0^1$

$\displaystyle V=\pi (\frac 12-\frac 13)$

$\displaystyle V=\frac 16\pi$

[snip]