# Triple integrals in spherical coordinates

• ACLerok
In summary, the speaker has a question about transforming triple integrals into spherical coordinates and is struggling to find the limits for the variable phi. They explain that phi is similar to theta and is usually integrated from 0 to Pi/2 for the top four octants and Pi/2 to Pi for the bottom four octants.
ACLerok
i have a question concerning transforming triple integrals into spherical coordinates. the problem is, i do not know how to find the limits of phi. Can anyone help me? Thanks...

What's the problem?

Phi is the variable indicating the angle subtended from the positive z-axis to the negative z-axis. It is similar to theta which is usually defined from positive x, but all the way around to positive x again.

Phi is usually integrated from 0 to Pi/2 for the top four octants, and Pi/2 to Pi for the bottom four octants. Anything more than that and we'd need a problem.

## 1. What are triple integrals in spherical coordinates?

Triple integrals in spherical coordinates are a mathematical tool used to calculate the volume of a three-dimensional object that is described in terms of spherical coordinates. It involves integrating over three variables: radius, azimuth angle, and inclination angle.

## 2. How do you convert Cartesian coordinates to spherical coordinates?

To convert from Cartesian coordinates (x, y, z) to spherical coordinates (r, θ, φ), you can use the following formulas: r = √(x² + y² + z²), θ = arctan(y/x), and φ = arccos(z/r). Alternatively, you can use online conversion tools or a graphing calculator.

## 3. What is the difference between spherical and Cartesian coordinates?

Spherical coordinates use a different system of coordinates than Cartesian coordinates. Instead of using x, y, and z coordinates, spherical coordinates use r, θ, and φ. These represent the distance from the origin, the angle from the positive x-axis, and the angle from the positive z-axis, respectively.

## 4. When is it useful to use triple integrals in spherical coordinates?

Triple integrals in spherical coordinates are useful when dealing with objects that have spherical symmetry, such as spheres, cones, or cylinders. They can also be more convenient to use when the shape of the object is better described in terms of spherical coordinates.

## 5. How do you set up a triple integral in spherical coordinates?

To set up a triple integral in spherical coordinates, you will need to determine the limits of integration for each variable (r, θ, and φ) based on the shape of the object. These limits will then be used in the integral expression: ∫∫∫f(r, θ, φ) dr dθ dφ. The order in which you integrate the variables may also vary depending on the problem.

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