Volume Integral of f over Sphere: Find Solution

• kidsmoker
In summary, the problem is to find the volume integral of the function f=x^2+y^2+z^2 over the region inside a sphere of radius R, centered on the origin. Using spherical polars, the triple integral can be simplified to: V=\int^{R}_{0}dr\int^{\pi/2}_{-\pi/2}d\theta\int^{2\pi}_{0}d\phi (r^{4}sin\theta). The limits of integration for theta are [-pi/2,pi/2]. The integrand can also be simplified to r^4sin\theta. The mistake of calling it a "surface integral" is due to the person being tired.
kidsmoker

Homework Statement

Find the volume integral of the function $$f=x^{2}+y^{2}+z^{2}$$ over the region inside a sphere of radius R, centered on the origin.

Homework Equations

Spherical polars $$x=rsin(\theta)cos(\phi), y=rsin(\theta)sin(\phi), z=rcos(\theta)$$

Jacobian in spherical polars = $$r^2sin(\theta)$$

The Attempt at a Solution

When i work through it I end up with the triple integral

$$V=\int^{R}_{0}dr\int^{\pi}_{-\pi}d\phi\int^{\pi}_{-\pi}d\theta (r^{2}sin^{2}\theta cos^{2}\phi+r^{2}sin^{2}\theta sin^{2}\phi + r^{2}cos^{2}\theta)r^2sin\theta$$

but I'm not too sure whether this is right. Mainly I'm not sure about the limits of integration.

Thanks.

If you want to integrate over all theta, that's only theta in [-pi/2,pi/2], isn't it? And you can simplify the integrand a LOT. x^2+y^2+z^2=r^2.

Ah yeah I didn't bother simplifying the integrand but I can see I should have done cos it would have made it a lot easier to type lol. I thought about it some more and understand why the limits are as you said now. Thanks!

Is there are reason why you titled this "surface integral"?

Because I was tired :D Oops lol

1. What is a volume integral?

A volume integral is a mathematical concept used to calculate the total volume of a three-dimensional region. It involves integrating a function over the desired region to find the volume.

2. How is the volume integral of f over a sphere calculated?

The volume integral of f over a sphere is calculated by integrating the function f over the entire surface of the sphere. This can be done using spherical coordinates and the formula for the volume of a sphere.

3. What is the significance of finding the volume integral of f over a sphere?

Finding the volume integral of f over a sphere can be used to solve a variety of physical and mathematical problems. It can be used to calculate the volume of a spherical object or to find the mass or charge distribution within a spherical region.

4. Are there any special techniques for solving volume integrals over spheres?

Yes, there are special techniques such as using spherical coordinates and symmetry arguments to simplify the integration process. It is also helpful to use the formula for the volume of a sphere and to break the integral into smaller parts.

5. Can the volume integral of f over a sphere be negative?

Yes, the volume integral of f over a sphere can be negative if the function f has negative values over certain regions of the sphere. This indicates that there is a net outflow of the function from the sphere.

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