Show that a triple integral = pi/4

In summary, to solve this improper integral, we need to convert to spherical coordinates and integrate over the first octant. This results in the integral becoming \int\int\int ρ^{3} e^{-ρ^{2}} sin\phi d\rho d\phi d\theta, with the bounds of \theta ranging from 0 to \pi/2, \phi ranging from 0 to \pi/2, and \rho ranging from 0 to infinity.
  • #1
hm8
16
0

Homework Statement



Show that

[itex]\int\int\int \sqrt{x^{2}+y^{2}+z^{2}} [/itex] [itex] e^{-({x^{2}+y^{2}+z^{2}})} dxdydz = \pi/4[/itex] where the bounds of x, y, and z are 0 to infinity

(The improper integral is defined as the limit of a triple integral over the piece of a solid sphere which lies in the first octant as the radius of the sphere increases indefinitely).

Homework Equations



In spherical coordinates, ρ2 = x2 + y2 + z2
dxdydz = ρ2sin∅ drho dpho dtheta

The Attempt at a Solution



I tried converting to spherical coordinates, which gave me

[itex]\int\int\int ρ^{3} [/itex] [itex] e^{-ρ^{2}} sin\phi d\rho d\phi d\theta[/itex]

But I'm not sure what my bounds would be (isn't ρ in relation to theta or phi somehow?) or even if I did, I'm not sure I could integrate it...
 
Physics news on Phys.org
  • #2
Since x, y, z range from 0 to infinity, you are talking about the first octant. To cover all four quadrants, [itex]\theta[/itex] normally ranges from 0 to [itex]2\pi[/itex]. To cover just the first quadrant, it ranges from 0 to [itex]\pi/2[/itex]. To cover all values of z, [itex]\phi[/itex] normally ranges from 0 to [itex]\pi[/itex], to cover just z> 0 it must vary from 0 to [itex]\pi/2[/itex].
 
  • #3
What about p?
 

FAQ: Show that a triple integral = pi/4

1. What is a triple integral?

A triple integral is a mathematical concept used in calculus to calculate the volume of a three-dimensional object. It involves integrating a function over a three-dimensional region.

2. How do you set up a triple integral?

A triple integral is typically set up using three nested integrals, with each integral representing a different dimension. The innermost integral represents the integration with respect to the x-variable, the middle integral represents the integration with respect to the y-variable, and the outermost integral represents the integration with respect to the z-variable.

3. What is the purpose of showing that a triple integral equals pi/4?

Showing that a triple integral equals pi/4 is a way to demonstrate the use of triple integrals in calculating the volume of a three-dimensional object. It also illustrates the application of mathematical concepts such as spherical coordinates and trigonometric functions.

4. Can you explain the process of solving a triple integral?

To solve a triple integral, first determine the limits of integration for each variable. Then, apply the appropriate integration techniques, such as substitution or integration by parts, to evaluate the integrals. Finally, multiply the results of the three integrals together to obtain the final value.

5. How can I check my answer to a triple integral problem?

You can check your answer to a triple integral problem by using a graphing calculator or software to plot the 3D region and comparing it to the calculated volume. You can also try solving the problem using a different method to see if you get the same result.

Back
Top