# Volume of a region

1. Nov 19, 2007

### stunner5000pt

1. The problem statement, all variables and given/known data
Find the volume of the region bounded by the surface given in the figure.
Until the image is approved this description will have to do.
What is the volume enclosed by the X-Y plane, a cylinder of radius r, and a plane that passes through the center of the cylinder and makes an angle theta with the X-Y plane?

(Hope that was clear enough)

2. The attempt at a solution
Ok since i see a cylinder i think cylindrical coordinates
Is this the correct answer in cylindrical coordinates? Ok i need this problem reformulated in Cartesian because this is for a friend in first year calculus

$$V = \int_{0}^{r} s ds \int_{0}^{\pi} d\phi \int_{0}^{r\cos\theta} dz$$

The reason there is a pi because this is only sweeping half circle. There is a $z\cos \theta$ as upper limit in the z integral because the length of the z varies between zero and $r\cos\theta$.

Is this correct? In cylindrical coordinates? If it is then i can proceed to do it through method of surfaces of revolution in the Cartesian system.

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Last edited: Nov 20, 2007
2. Nov 20, 2007

### stunner5000pt

i have revised my solution. The phi integrand was wrong.

I just need someone to tell me if im right or wrong. Thanks a lot for your help!

3. Nov 20, 2007

### stunner5000pt

bump :)

4. Nov 21, 2007

### HallsofIvy

Staff Emeritus
bump yourself! (I really hate that! If people want to answer your question, they will!)

"What is the volume enclosed by the X-Y plane, a cylinder of radius r, and a plane that passes through the center of the cylinder and makes an angle theta with the X-Y plane?"
passes through the center of the cylinder? Do you mean "Passes through the center of the base of the cylinder in the xy-plane"? That's what your picture shows.
If that is the case, then it looks to me like your integral is correct.

5. Nov 21, 2007

### stunner5000pt

now thats waht i thought too and the answer i get yields
$$\frac{r^2}{2} * \pi * r\cos\theta$$

but the answer in the textbook is
$$\frac{2}{3}r^3 \cos\theta$$

6. Nov 23, 2007

### alpha372

What book are you using?