Triple Integrals in Spherical Coordinates

[batch3]

Hi all,

I'm not sure how to get the boundaries in terms of both the spherical and cylindrical coordinates for this question.

View attachment 2504

Here are the boundaries we were given in the solution.

View attachment 2505

How was \(\displaystyle \frac{\pi}{4}\) for φ and \(\displaystyle \frac{1}{\sqrt{2}}\) for r obtained?

Thanks!

 

[jvicens]

Hi there,

In order to understand how the boundaries were obtained, it's important to first understand the coordinate systems being used. In spherical coordinates, a point is represented by its distance from the origin (r), the angle formed by the point and the positive z-axis (θ), and the angle formed by the point and the positive x-axis (φ). In cylindrical coordinates, a point is represented by its distance from the origin (ρ), the angle formed by the point and the positive z-axis (θ), and the height above the xy-plane (z).

Now, for the boundaries in question, we need to consider the given solution. The problem likely involves a specific region or volume in space, and the boundaries given are the limits for the variables r and φ. In this case, it seems that the region is a quarter of a sphere, with the angle φ ranging from 0 to π/2. This is because a quarter of a sphere has a maximum angle of π/2, since the other three quarters are just reflections of this one.

As for the value of r, it is likely related to the radius of the sphere that the region is a part of. For example, if the sphere has a radius of 1, then the maximum value of r would be 1, since any point outside of the sphere would not be included in the region. In this case, the value of r is given as 1/√2, which is likely the radius of the sphere or a related value.

I hope this helps explain the boundaries and how they were obtained. Let me know if you have any further questions.