CPM Homework Help : APCALC Problem 8-51

Calculate the volume of the region bounded by $y =\sqrt { x }$ , $y = 0$ , and $x = 4$ revolved about each of the lines below. Use the steps outlined in problem 8-49.

$x$ -axis
Hint (a):
After sketching an arbitrary rectangle (with $Δx→0$ ), 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.
$y$ -axis
Hint 1(b):
This arbitrary rectangle indicates that there will be a hole in the center of this solid. You will have to use the Washer Method.
Hint 2(b):
Horizontal rectangles indicate that the bounds and the integrand will be in terms of $y$ .
Hint 3(b):
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 $r = y^2$ .
$x = −2$
Hint 1(c):
Q: Washers or disks? Think: Will there be a hole or won't there?
Hint 2(c):
Q: Will the integral be in terms of $x$ or $y$ ? Think: Are the rectangles vertical or horizontal?
Hint 3(c):
Notice that these radii are $2$ units longer than the radii in part (b).
$R = 2+4$
$r = 2 + y^2$
$y = 5$
Hint (d):
$R = 5$
Hint (d):
Since $y=\sqrt{x}$ represents the distance between the $x\text{-axis and }y=\sqrt{x}$ , it follows that: $r=5-\sqrt{x}$

First quadrant, increasing concave down curve, labeled y =square root of x, starting at the origin & ending @ (4, comma 2), with vertical segment from (4, comma 0) to (4, comma 2), region below curve, above x axis, & left of x = 4, shaded.