# Cylinder in 3 D

1. Dec 13, 2017

### songoku

1. The problem statement, all variables and given/known data
Let r be a positive constant. Consider the cylinder x2 + y2 ≤ r2, and let C be the part of the cylinder that satisfies
0 ≤ z ≤ y.
(1) Consider the cross section of C by the plane x = t (−r ≤ t ≤ r), and express its area in terms of r, t.
(2) Calculate the volume of C, and express it in terms of r.
(3) Let a be the length of the arc along the base circle of C from the point (r, 0, 0) to the point (r cos θ, r sin θ, 0)
(0 ≤ θ ≤ π). Let b be the length of the line segment from the point (r cos θ, r sin θ, 0) to
the point (r cos θ, r sin θ, r sin θ). Express a and b in terms of r, θ.
(4) Calculate the area of the side of C with x2+y2 = r2, and express it in terms of r.

2. Relevant equations
Not sure

3. The attempt at a solution
Let the x - axis horizontal, y - axis vertical and z - axis in / out of page. I imagine there is circle on xy plane with radius r then it extends out of page (I take out of page as z+) to form 3 D cylinder.

(1) I image there is rectangular plane that cuts the cylinder and the shape of the cross section is rectangle. The length of the triangle is 2y and the width is z. Taking z = y, the area will be 2y2 = 2 (r2 - x2) = 2 (r2 - t2)
But the answer is 1/2 (r2 - t2)

(2) The volume of cylinder = base area x height = πr2 . z and by taking z = y = r I get πr3 but the answer is 2/3 r3

(3) I get this part

(4) I am not sure what "area of side of C" is. Is it surface area of the circular part of cylinder?

Thanks

2. Dec 13, 2017

### BvU

Nope. Make a sketch.

3. Dec 13, 2017

### songoku

I did. I made vertical line that cuts x - axis then stretched it in z - axis and made the plane cut the cylinder and the cross section of the intersection looked like rectangle. Or maybe I don't know which part called "cross - section"

4. Dec 13, 2017

### LCKurtz

The plane $z=y$ is a $45^\circ$ plane that cuts through your cylinder at a slant. The portion described makes a wedge and you want cross sections parallel to the $zy$ plane because the plane $x=t$ for fixed $t$ is parallel to the $zy$ plane.

5. Dec 13, 2017

### BvU

Simple case: x = 0. So -1 ≤ y ≤ 1. In the yz plane 0 ≤ z ≤ y is a triangle.

6. Dec 13, 2017

### songoku

I think I am missing something here because I feel I can't really grasp the hint given.

Let me start from the basic again:
1. Let the x - axis horizontal, y - axis vertical and z - axis in / out of page. I imagine there is circle on xy plane with radius r then it extends out of page (I take out of page as z+) to form 3 D cylinder. Is this correct?

2. Plane x = t is like the shape of a piece of paper hold vertically with the face of paper facing x - axis (I mean x - axis is the normal of the plane). Is this correct?

Thanks

7. Dec 14, 2017

### BvU

Yes

8. Dec 14, 2017

### songoku

"Consider the cross section of C by plane x = t" means plane x = t cuts the cylinder?

And the intersection will be rectangle?

9. Dec 14, 2017

### BvU

yes. Infinitely extending in the + and -z direction.

Now what about the part of the cylinder that satisfies 0 ≤ z ≤ y ?

10. Dec 14, 2017

### songoku

Ahh I think I am starting to see the direction of the hint.

The intersection of plane x = t and C is in the shape of right angle triangle with its base and height equal to y so the area will be 1/2 y2 = 1/2 (r2 - t2). Is this correct?

For (2), should I use integration to find the volume?
$$\int_{0}^{r} y^{2} dx$$
$$= \int_{0}^{r} (r^{2} - x^{2} dx$$
$$= \frac{2r^{3}}{3}$$

Or maybe there is non - calculus way?

For (4), does the question ask to find the surface area of C?

Thanks

11. Dec 15, 2017

### BvU

you are on the right track! is it luck or did you deal with the 1/2 by integrating from 0 to r instead of from -r to r ?

for 4: only what is on the surface of the cylinder

12. Dec 16, 2017

### songoku

Yes I deal with it as you say
In my mind, I will get the area by integrating z dx and because z = y, it is the same as integrating (r2 - x2) dx but I get different answer from the answer key. So it is not correct integrating z dx to find the surface area?

13. Dec 16, 2017

### LCKurtz

No, that is not correct. Your exercise is trying to get you to work with $\theta$ and $z$ variables (cylindrical coordinates). What is the element of length along the circle in the $xy$ plane? What is the element of length in the $z$ direction? You can use them to build the surface area element $dS$. You haven't indicated what you have studied, but if you have studied parametric representations of surface areas you can get $dS$ from that too.

14. Dec 16, 2017

### Staff: Mentor

Thread moved. @songoku, please post problems that involve integrals in the Calculus & Beyond section.

15. Jan 9, 2018