## monroe17 Group Title 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 one year ago one year ago

1. NoelGreco Group Title

What are the points of intersection?

2. yakeyglee Group Title

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. $V=2\pi \int_a^b x (f-g) dx$

3. monroe17 Group Title

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

4. NoelGreco Group Title

|dw:1361312140525:dw| 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.

5. yakeyglee Group Title

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?

6. yakeyglee Group Title

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

7. monroe17 Group Title

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

8. yakeyglee Group Title

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

9. monroe17 Group Title

what defines an upper and lower bound?

10. NoelGreco Group Title

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

11. yakeyglee Group Title

@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).

12. NoelGreco Group Title

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

13. NoelGreco Group Title

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

14. yakeyglee Group Title

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