In my calculus course, we just covered the Shell Method and its uses. I have been doing the homework for a few hours and I am absolutely stumped by a question.
The question states:
Find the volume of the solid generated by revolving the region in the first quadrant bounded by the graph of the equation about the given line.
$$ x^{2/3} + y^{2/3} = 12^{2/3} $$
(i) the $x$-axis; (ii) the $y$-axis
The question is multiple choice and lists the following possible answers:
- (i) $\frac{13824\pi}{25}$; (ii) $\frac{13824\pi}{25}$
- (i) $\frac{3456\pi}{25}$; (ii) $\frac{18432\pi}{35}$
- (i) $\frac{4608\pi}{35}$; (ii) $\frac{18432\pi}{35}$
- (i) $\frac{18432\pi}{35}$; (ii) $\frac{18432\pi}{35}$
- (i) $\frac{18432\pi}{35}$; (ii) $\frac{4608\pi}{35}$
I quickly worked out the answer to the first part (the $x$-axis) and I ended up with $$\frac{18432\pi}{35}$$
This all seemed fine to me and it narrowed my options down to either choice 4 or 5.
However, once I started working on the second part (the $y$-axis) things got messy.
When I solved for $y$, I ended up with:
$$y = \sqrt{[12^{2/3} - x^{2/3}]^3}$$
This could of course be simplified further, but I had no problem with this equation. It seemed correct and I felt ready for the next step.
Because this is the Shell Method about the y-axis, I needed to find the information for a rectangle which ran parallel to $x = 0$. So I drew up a graph, added a rectangle, and began labeling it (for the sake of keeping track of things). Goal number 1: find $h(x)$. At any given value, the height of my parallel rectangle should be a point on the curve. So this led me to say:
$$h(x) = \sqrt{[12^{2/3} - x^{2/3}]^3}$$
Next I needed to identify $p(x)$. This was also fairly. Because I am using the y-axis as my axis of revolution $p(x)$ should just be:
$$p(x) = x$$
Now the shell method defines the equation for a vertical axis of revolution as:
$$ V = 2\pi \int_a^b p(x)h(x)dx $$
So filling everything in:
$$ V = 2\pi \int_0^{12} (x \sqrt{[12^{2/3} - x^{2/3}]^3})dx $$
The definite integral of this produced $\frac{9216\pi}{35}$. However, not only do I not have this option:
(i) $\frac{18432\pi}{35}$; (ii) $\frac{9216\pi}{35}$
The question does not give any option which includes "$\frac{9216\pi}{35}$".
Can anybody help me identify what I did wrong and how I can fix it? I would be very grateful.
I have also included a screenshot of my work: (Please excuse my poor handwriting in the image. I have only recently begun using a tablet for working out problems.)
In the image, "3/2" is used rather than the square root of the cube.