Calculating 3D Cylinder Volumes & Areas With Constants

[songoku]

Homework Statement

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.

Homework Equations

Not sure

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

 

[BvU]

I image there is rectangular plane that cuts the cylinder and the shape of the cross section is rectangle

Nope. Make a sketch.

 

[songoku]

Nope. Make a sketch.

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"

 

[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.

 

[BvU]

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.

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

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"

What about y ?

 

[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

 

[BvU]

Yes

 

[songoku]

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

And the intersection will be rectangle?

 

[BvU]

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

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

 

[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

 

[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

 

[songoku]

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 ?

Yes I deal with it as you say

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

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?

 

[LCKurtz]

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?

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.

 

[Mark44]

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

 

[songoku]

Sorry for really late reply

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

I am really sorry. I don't know this question involves integration. Thanks a lot for the help

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.

I'll try to think about your hint then replying back.

Thanks