Integrating (triple) over spherical coordinates

In summary: So the limits of your integrals should be:\int_0^{\frac{\pi}{2}} \int_0^{\frac{\pi}{2}} \int_{1}^{2} r\ dz\ dr\ d\thetaIn summary, the problem calls for finding the volume bounded by two hemispheres in the first octant using spherical coordinates. The volume element is given by dV=r^2 * sin(theta) * dr * d(phi) * d(theta), and the limits of integration are 0 to pi/2 for both theta and phi, and 1 to 2 for r.
  • #1
MorallyObtuse
45
0
Hi,

Set up the triple integral in spherical coordinates to find the volume bounded by \(\displaystyle z = \sqrt{4-x^2-y^2}\), \(\displaystyle z=\sqrt{1-x^2-y^2}\), where \(\displaystyle x \ge 0\) and \(\displaystyle y \ge 0\).

\(\displaystyle \int_0^{2\pi} \int_0^2 \int_{-\sqrt{4-x^2-y^2}}^{\sqrt{4-x^2-y^2}} r\ dz\ dr\ d\theta\)
 
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  • #2
I don't think you're using spherical coordinates yet - your integrals look more like cylindrical. This problem could probably be done either way, but the problem did say to use spherical. So, assuming that $\varphi$ is the azimuthal angle, and $\theta$ is the polar angle (some authors switch these two), your volume element is given by
$$dV=r^2 \sin(\theta) \, dr \, d\varphi \, d\theta.$$
Can you continue from here?
 
  • #3
Taylor Kane said:
Hi,

Set up the triple integral in spherical coordinates to find the volume bounded by \(\displaystyle z = \sqrt{4-x^2-y^2}\), \(\displaystyle z=\sqrt{1-x^2-y^2}\), where \(\displaystyle x \ge 0\) and \(\displaystyle y \ge 0\).

\(\displaystyle \int_0^{2\pi} \int_0^2 \int_{-\sqrt{4-x^2-y^2}}^{\sqrt{4-x^2-y^2}} r\ dz\ dr\ d\theta\)

You should also note that both of the bounding solids are hemispheres centred at the origin, the top being of radius 2 and the bottom of radius 1.

But because we are told x >= 0 and y >= 0, we are only going to be in the first octant.

That means your longitudinal angles do NOT sweep out a full circle, they will only sweep out a quarter circle. Your latitudinal angles only sweep out the top half.
 

FAQ: Integrating (triple) over spherical coordinates

What is the purpose of integrating over spherical coordinates?

Integrating over spherical coordinates allows us to solve problems in three-dimensional space, where traditional two-dimensional coordinate systems are inadequate.

What are the three variables used in spherical coordinates?

The three variables used in spherical coordinates are radius (r), polar angle (θ), and azimuthal angle (φ).

How do you convert from Cartesian coordinates to spherical coordinates?

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

What is the range of values for each variable in spherical coordinates?

The radius (r) can range from 0 to infinity, the polar angle (θ) can range from 0 to π, and the azimuthal angle (φ) can range from 0 to 2π.

What are some real-world applications of integrating over spherical coordinates?

Spherical coordinates are commonly used in physics, engineering, and astronomy to describe the motion and positions of objects in three-dimensional space. They are also useful in solving problems involving electric and magnetic fields, as well as in calculating the volume and surface area of three-dimensional objects.

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