Question

Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by y=3sqrt(x) and y=3x^2 about the y-axis

Answer 1

What are the points of intersection?

Answer 2

For two functions \(f\) and \(g\) where \(f \ge g\), the volume between \(x=a\) and \(x=b\) by the cylindrical shells method is given by the following formula. You find those bounds by finding the points of intersection of \(f\) and \(g\), presumably.
\[V=2\pi \int_a^b x (f-g) dx\]

Answer 3

0 and 1 are the points of intersection.. but I can't seem to figure out the function to evaluate the integral at ;/

Answer 4

Each shell is going to have the shape of a rectangle when it is unfolded from the circular shell shape. You are really looking for the volume of that rectangle.

V = length X height X thickness

The thickness is going to be dx or dy, the height is the difference between the two functions, and the length is the circumference of each shell.

Answer 5

Using the formula, we have \(f=3\sqrt x\) and \(g=3x^2\). So\[V=2\pi \int_a^b x (3 \sqrt x + 3x^2) dx.\]You can simplify and integrate that, yes?

Answer 6

I should have put \(\displaystyle \int_0^1\), but you get the idea.

Answer 7

how did you know what f and g were..? do you just assume?

Answer 8

They were the equations you were given (\(f\) is the upper bound and \(g\) is the lower bound). Does that make sense?

Answer 9

what defines an upper and lower bound?

Answer 10

I think you want the DIFFERENCE between the two functions in the integrand.

Answer 11

@NoelGreco, I do. Look at the integral again.

@monroe17, the upper bound is the function that defines the top part of the bounded area, and the lower bound defines the lower part of the bounded area (look at their positions in the picture you drew).

Answer 12

I looked at both. You have the sum, not the difference.

Answer 13

Just make sure you get \[\frac{ 9\Pi }{ 10 }\]

Answer 14

That is what I meant.
\[V=2\pi \int_\color{red}{0}^\color{red}{1} x (3 \sqrt x \color{red}{-} 3x^2) dx.\]