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8-51.

Using the methods you have learned, find the volume of the region bounded by: $y=\sqrt{x}$, $y = 0$, and $x = 4$, revolved about the lines below. Use the steps outlined in problem 8-49. 1. $x$-axis

After sketching an arbitrary rectangle (with $\Delta x→0$), it is apparent that there will not NOT be a hole in the center of this solid.
That indicates that the Disk Method will be appropriate.

2. $y$-axis

This arbitrary rectangle indicates that there will be a hole in the center of this solid. You will have to use the Washer Method.

Horizontal rectangles indicate that the bounds and the integrand will be in terms of $y$.

Notice that the figure will be a cylinder with a curvy hole in the center.
The radii of the cylinder is $R = 4$.
The radii of the curvy hole is $r=y^2$.

3. $x = −2$

Q: Washers or disks? Think: Will there be a hole or won't there?

Q: Will the integral be in terms of $x$ or $y$? Think: Are the rectangles vertical or horizontal?

Notice that these radii are $2$ units longer than the radii in part (b).
$R = 2+4$
$r=2+y^2$

4. $y = 5$

$R = 5$

$\text{Since }y=\sqrt{x}\text{ represents the distance between the }x\text{-axis and }y=\sqrt{x},\text{ it follows that: }r=5-\sqrt{x}$