# How do you evaluate a triple integral using spherical coordinates?

## How do you evaluate a triple integral using spherical coordinates?

To evaluate a triple integral in spherical coordinates, use the iterated integral ∫θ=βθ=α∫ρ=g2(θ)ρ=g1(θ)∫u2(r,θ)φ=u1(r,θ)f(ρ,θ,φ)ρ2sinφdφdρdθ.

### How do you use spherical coordinates?

To convert a point from spherical coordinates to cylindrical coordinates, use equations r=ρsinφ,θ=θ, and z=ρcosφ. To convert a point from cylindrical coordinates to spherical coordinates, use equations ρ=√r2+z2,θ=θ, and φ=arccos(z√r2+z2).

#### How do you find the volume of a spherical coordinate?

Use spherical coordinates to find the volume of the triple integral, where B is a sphere with center ( 0 , 0 , 0 ) (0,0,0) (0,0,0) and radius 4. Using the conversion formula ρ 2 = x 2 + y 2 + z 2 \rho^2=x^2+y^2+z^2 ρ2​=x2​+y2​+z2​, we can change the given function into spherical notation.

How to calculate triple integrals in spherical coordinates?

Example 1 Evaluate ∭ E 16zdV ∭ E 16 z d V where E E is the upper half of the sphere x2 +y2 +z2 =1 x 2 + y 2 + z 2 = 1 . Example 2 Evaluate ∭ E zxdV ∭ E z x d V where E E is above x2 +y2 +z2 = 4 x 2 + y 2 + z 2 = 4, inside the cone (pointing upward) that makes an angle of π 3 π 3 with the negative z z -axis and has x ≤ 0 x ≤ 0 .

Which is an example of a triple integral?

Now that we have the limits we can evaluate the integral. Example 3 Convert ∫ 3 0 ∫ √9−y2 0 ∫ √18−x2−y2 √x2+y2 x2 +y2 +z2dzdxdy ∫ 0 3 ∫ 0 9 − y 2 ∫ x 2 + y 2 18 − x 2 − y 2 x 2 + y 2 + z 2 d z d x d y into spherical coordinates. Let’s first write down the limits for the variables.

## What are the restrictions on the coordinates of an integral?

We also have the following restrictions on the coordinates. For our integrals we are going to restrict E E down to a spherical wedge. This will mean that we are going to take ranges for the variables as follows, Here is a quick sketch of a spherical wedge in which the lower limit for both ρ ρ and φ φ are zero for reference purposes.

### Which is the triple integral for a circular cylinder?

This means that the circular cylinder x2 + y2 = c2 in rectangular coordinates can be represented simply as r = c in cylindrical coordinates. (Refer to Cylindrical and Spherical Coordinates for more review.) Triple integrals can often be more readily evaluated by using cylindrical coordinates instead of rectangular coordinates.