1. ## cylindrical coordinate problem

find the volume bounded by cylinder (y^2 0 + (z^2 0 =1 , plane y = x and y-z plane in first octant...

I'm not sure to let my y = rcos theta or r sin (theta) , but i do know that volume must be positive....
I'm not convinced that why should i use y = r sintheta

2. ## Re: cylindrical coordinate problem

Hey xl5899.

I should ask what are three variables you are using and what is your co-ordinate system representation [in terms of normal Cartesian and/or standardized system]?

It's a bit difficult to tell from your integral and picture since they seem to use different systems.

3. ## Re: cylindrical coordinate problem

Originally Posted by xl5899
find the volume bounded by cylinder (y^2 0 + (z^2 0 =1 , plane y = x and y-z plane in first octant...

I think you mean "y^2+ z^2= 1". I am not clear what the two "0" s mean. And I think you mean the planes "y= x" and "y= z".
One "usually" sees the cylinder given as x^2+ y^2= 1. That is a cylinder with axis the z-axis and whose cross section at the xy-plane is the circle centered at (0,0,0) and radius 1. That region can be parameterized by $\displaystyle x=r cos(\theta)$, $\displaystyle y= r sin(\theta)$. z= t. So it should be clear that, here, the rolls of "x" and "z" have been swapped. This is a cylinder with axis the x-axis and whose cross section at the yz-plane is the circle centered at (0, 0, 0) and radius 1.

I'm not sure to let my y = rcos theta or r sin (theta) , but i do know that volume must be positive....
I'm not convinced that why should i use y = r sintheta
Actually, what you are concerned with here is irrelevant! You can parameterize a circle with $\displaystyle x= r cos(theta)$ and $\displaystyle y= r sin(\theta)$ or $\displaystyle x= r sin(\theta)$ and $\displaystyle y= r cos(\theta)$. The only difference is that in the first $\displaystyle \theta= 0$ is the point (1, 0) while, in the second, it is (0, 1). But as $\displaystyle \theta$ goes from 0 to $\displaystyle 2\pi$, they both give the same circle.
The crucial point in this problem is that "x" and "z" have been reversed.
(You could get exactly the same answer by writing the problem as "find the volume bounded by cylinder x^2 + y^2 =1 , plane y = z and y= x plane in first octant...", by reversing "x" and "z" in the problem.)

4. ## Re: cylindrical coordinate problem

Originally Posted by HallsofIvy
I think you mean "y^2+ z^2= 1". I am not clear what the two "0" s mean. And I think you mean the planes "y= x" and "y= z".
One "usually" sees the cylinder given as x^2+ y^2= 1. That is a cylinder with axis the z-axis and whose cross section at the xy-plane is the circle centered at (0,0,0) and radius 1. That region can be parameterized by $\displaystyle x=r cos(\theta)$, $\displaystyle y= r sin(\theta)$. z= t. So it should be clear that, here, the rolls of "x" and "z" have been swapped. This is a cylinder with axis the x-axis and whose cross section at the yz-plane is the circle centered at (0, 0, 0) and radius 1.

Actually, what you are concerned with here is irrelevant! You can parameterize a circle with $\displaystyle x= r cos(theta)$ and $\displaystyle y= r sin(\theta)$ or $\displaystyle x= r sin(\theta)$ and $\displaystyle y= r cos(\theta)$. The only difference is that in the first $\displaystyle \theta= 0$ is the point (1, 0) while, in the second, it is (0, 1). But as $\displaystyle \theta$ goes from 0 to $\displaystyle 2\pi$, they both give the same circle.
The crucial point in this problem is that "x" and "z" have been reversed.
(You could get exactly the same answer by writing the problem as "find the volume bounded by cylinder x^2 + y^2 =1 , plane y = z and y= x plane in first octant...", by reversing "x" and "z" in the problem.)
so , due to the role of z and x has been reversed , we just sticked y = r sin theta in this problem ?

5. ## Re: cylindrical coordinate problem

Originally Posted by HallsofIvy
I think you mean "y^2+ z^2= 1". I am not clear what the two "0" s mean. And I think you mean the planes "y= x" and "y= z".
One "usually" sees the cylinder given as x^2+ y^2= 1. That is a cylinder with axis the z-axis and whose cross section at the xy-plane is the circle centered at (0,0,0) and radius 1. That region can be parameterized by $\displaystyle x=r cos(\theta)$, $\displaystyle y= r sin(\theta)$. z= t. So it should be clear that, here, the rolls of "x" and "z" have been swapped. This is a cylinder with axis the x-axis and whose cross section at the yz-plane is the circle centered at (0, 0, 0) and radius 1.

Actually, what you are concerned with here is irrelevant! You can parameterize a circle with $\displaystyle x= r cos(theta)$ and $\displaystyle y= r sin(\theta)$ or $\displaystyle x= r sin(\theta)$ and $\displaystyle y= r cos(\theta)$. The only difference is that in the first $\displaystyle \theta= 0$ is the point (1, 0) while, in the second, it is (0, 1). But as $\displaystyle \theta$ goes from 0 to $\displaystyle 2\pi$, they both give the same circle.
The crucial point in this problem is that "x" and "z" have been reversed.
(You could get exactly the same answer by writing the problem as "find the volume bounded by cylinder x^2 + y^2 =1 , plane y = z and y= x plane in first octant...", by reversing "x" and "z" in the problem.)
i mean find the volume bounded by cylinder x^2 + y^2 =1 , plane y-z and y= x plane in first octant...