Substituting spherical coordinates to evaluate an integral

In summary, the given problem involves evaluating a triple integral using spherical coordinates. The integral can be simplified to 3.57, but this may not be accurate as it does not match the expected value of 5.236. The volume of integration is a combination of a sphere for z<0 and a cylinder for z>0. It is possible that a different approach, such as using geometry, may be needed to accurately solve this problem.
  • #1
HMPARTICLE
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I have to evaluate

$$\int^1_{-1} \int^{ \sqrt {1-x^2}}_{ - \sqrt {1-x^2}} \int^1_{-\sqrt{x^2+y^2}}dzdydx$$

using spherical coordinates.

This is what I have come up with

$$\int^1_{0} \int^{ 2\pi}_0 \int^{3\pi/4}_{0}r^2\sin\theta d\phi d\theta dr$$

by a combination of sketching and substituting spherical coordinates.

After evaluating I obtain this integral to equal 3.57.

where as the first one evaluates to 5.236.

These are so difficult :(
 
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  • #2
I am not convinced you did that right.
The volume of integration appears to be a sphere for z<0 and a cylinder for z>0. Your spherical integral doesn't look like that.

Are you allowed to use spherical integral for the lower half and cylindrical integral for the upper half? Or maybe just geometry...1/2 volume of unit sphere + volume of unit cylinder = 5.236.
 

What are spherical coordinates?

Spherical coordinates are a coordinate system used to describe the position of a point in three-dimensional space. They use the distance from a fixed point, an angle from a fixed axis, and an angle from a fixed plane to specify the location of a point.

Why would I need to use spherical coordinates to evaluate an integral?

Spherical coordinates are particularly useful when evaluating integrals that involve spherical symmetry. They can simplify the integration process and lead to more elegant solutions.

How do I convert from Cartesian coordinates to spherical coordinates?

To convert from Cartesian coordinates (x, y, z) to spherical coordinates (r, θ, φ), you can use the following equations:
r = √(x² + y² + z²)
θ = arctan(y/x)
φ = arccos(z/r)

What is the significance of the limits of integration in spherical coordinates?

The limits of integration in spherical coordinates correspond to the size and shape of the region being integrated. They are typically determined by the boundaries of the integral in Cartesian coordinates and can be represented in terms of the spherical coordinates.

Are there any limitations to using spherical coordinates to evaluate an integral?

While spherical coordinates can be useful for certain types of integrals, they may not be the best choice for all problems. It is important to consider the complexity and symmetry of the integrand when deciding whether to use spherical coordinates or another coordinate system.

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