How to set bounds in cylindrical coordinates analytically?

In summary, the conversation discusses evaluating an integral in cylindrical coordinates and how to set the bounds analytically. The poster on Mathematics Stack Exchange provided an explanation on how to transform the bounds, but there was some confusion about the z-coordinate varying from 0 to 6 rather than the y-coordinate. The conversation ends with a suggestion to forget about understanding the explanation on Stack Exchange and instead translate the bounds directly from the original integral.
  • #1
cwill53
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Homework Statement
Evaluate the following integral in cylindrical coordinates.
$$\int_0^6 \int_0^{\frac{\sqrt{2}}{2}}\int_x^{\sqrt{1-x^2}}e^{-x^2-y^2} \, dy \, dx \, dz$$

After attempting to set the bounds in cylindrical coordinates, I got
$$\int_0^6 \int_0^{\frac{\sqrt{2}}{2}}\int_{\rho \cos\varphi }^{\sqrt{1-\rho^2 \cos^2\varphi }}e^{-\rho ^2}d\varphi \rho \, d\rho \, dz$$

But I know this doesn't make sense. Can someone explain how to switch the bounds analytically? I don't understand how to transform the bounds.

I had posted this question on the Mathematics Stack Exchange:
https://math.stackexchange.com/questions/3898044/how-to-set-bounds-in-cylindrical-coordinates
Relevant Equations
$$\begin{cases}x=\rho \cos \phi \\
y= \rho \sin \phi \\
z=z'\end{cases}$$
I'm trying to evaluate the following integral in cylindrical coordinates.
$$\int_0^6 \int_0^{\frac{\sqrt{2}}{2}}\int_x^{\sqrt{1-x^2}}e^{-x^2-y^2} \, dy \, dx \, dz$$

After attempting to set the bounds in cylindrical coordinates, I got
$$\int_0^6 \int_0^{\frac{\sqrt{2}}{2}}\int_{\rho \cos\varphi }^{\sqrt{1-\rho^2 \cos^2\varphi }}e^{-\rho ^2}d\varphi \rho \, d\rho \, dz$$

But I know this doesn't make sense. Can someone explain how to switch the bounds analytically? I don't understand how to transform the bounds.

I had posted this question on the Mathematics Stack Exchange:
https://math.stackexchange.com/questions/3898044/how-to-set-bounds-in-cylindrical-coordinates

Screenshot 2020-11-07 at 4.15.50 PM.png


This is what one poster said. But the z-coordinate, not the y-coordinate, is what varies from 0 to 6.

So the inequality should be

$$\left\lbrace \begin{array}{a}
x\leqslant \rho \sin \phi \leqslant \sqrt{1-x^{2}}\\
0 \leqslant \rho \cos \phi \leqslant \frac{1}{\sqrt{2}}
\end{array} \right\rbrace$$
However I need some assistance solving the inequality so that it would help me set the bounds.
 
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  • #2
Draw an x-y plane and shade the area over which the integral is done
 
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  • #3
BvU said:
Draw an x-y plane and shade the area over which the integral is done
Can you explain how he solved the inequalities?
 
  • #4
I don't understand what he did either, how he extracted those limits from the inequality with the ##\text{min}()## in it. Maybe someone clever can explain, but I wouldn't worry about it.
 
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  • #5
Dear cwill,
Forget about understanding the nonsense in stackexchange.
$$\int_0^{\sqrt 2\over 2}\;\int_x^\sqrt{1-x^2}\ ... \ dy\, dx $$ Surely you can translate this to
##\qquad\qquad x## runs from 0 to ##{1\over 2}\sqrt 2##
##\qquad\qquad y## runs from ##x## to ## \sqrt{1-x^2}##

and, once you make a sketch, the translation to bounds in ##\rho## and ## \phi## is evident.
 
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1. What are cylindrical coordinates?

Cylindrical coordinates are a type of coordinate system used in mathematics and physics to describe the position of a point in three-dimensional space. They consist of a distance from the origin (r), an angle from the positive x-axis (θ), and a height or depth (z).

2. How do I convert from Cartesian coordinates to cylindrical coordinates?

To convert from Cartesian coordinates (x, y, z) to cylindrical coordinates (r, θ, z), you can use the following equations: r = √(x² + y²), θ = arctan(y/x), and z = z. These equations can also be used to convert from cylindrical coordinates to Cartesian coordinates.

3. How do I set bounds in cylindrical coordinates?

To set bounds in cylindrical coordinates, you need to define the range of values for each coordinate. For example, if you want to set bounds for r between 0 and 10, θ between 0 and 2π, and z between -5 and 5, your bounds would be 0 ≤ r ≤ 10, 0 ≤ θ ≤ 2π, and -5 ≤ z ≤ 5.

4. Can I set asymmetric bounds in cylindrical coordinates?

Yes, you can set asymmetric bounds in cylindrical coordinates. This means that the bounds for each coordinate do not have to be the same on both sides. For example, you can have 0 ≤ r ≤ 5 on one side and 0 ≤ r ≤ 10 on the other side.

5. How do I use cylindrical coordinates to solve problems in physics or engineering?

Cylindrical coordinates are particularly useful in solving problems involving cylindrical or rotational symmetry, such as in fluid dynamics or electromagnetism. They can also be used to simplify calculations in problems involving curved surfaces or objects. It is important to understand how to convert between cylindrical and Cartesian coordinates in order to effectively use them in problem-solving.

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