online class and I don't know what to do so I posts. Any help is great.Thank you
Convert the integral [0,1]∫ [0,√(1-x^2 -y^2)]∫𝑧√(x^2 +y^2 +z^2)dz dy dx into anequivalent integral in spherical coordinates and evaluate the integral.
Hmmm. I did mess up on that one; I didn't refer to my vector analysis text first. However, you only have a double integral, and 3 variables. I expect that the limits for y are [1,√(1-x^2)].
This one is similar to the polar coordinates one. The volume element
dv = dz dy dx becomes dv = r^2 sinθ dr dθ dØ
If that isn't clear, google volume element in spherical coordinates to see how this is done. Your class materials surely also derive the transformation.
Now, with z = r cosθ
the integrand z √(x^2 +y^2 +z^2) = r cosθ * r = r^2 cosθ and so the integral becomes ∫∫∫ r^2 cosθ * r^2 sinθ dr dθ dØ
and the region appears to be the 1st octant of the sphere of radius 1, so the limits on θ and Ø should be pretty easy. See what you can come up with.
You have ∫∫∫ r^2 sinθ cosØ dA
so change the integration limits
How did you get r^2 sinθ cosØ dA. Can you show a some sample of how you would set them up. I don't understand with just ∫∫∫ r^2 sinθ cosØ dA for the question.
Convert the integral [0,1]∫ [0, √(1-x^2)∫ [0,√(1-x^2 -y^2)]∫𝑧√(x^2 +y^2 +z^2)dz dy dx into an equivalent integral in spherical coordinates and evaluate the integral.
To convert the given integral to spherical coordinates, we need to express the limits of integration and the differential volume element in terms of spherical coordinates.
In spherical coordinates, we have three variables: ρ (rho), θ (theta), and φ (phi).
ρ represents the distance from the origin to the point in space, θ represents the angle with the positive x-axis in the xy-plane, and φ represents the angle with the positive z-axis.
To determine the limits of integration for ρ, θ, and φ, we analyze the given region of integration.
The given integral has [0, 1] as the limits of integration for x, which means ρ varies from 0 to 1. Similarly, y has limits [0, √(1-x^2 -y^2)]. In spherical coordinates, ρ can vary from 0 to 1, θ can vary from 0 to 2π (for a complete rotation in the xy-plane), and φ can vary from 0 to π/2 (since the given region is in the positive z-direction).
So, the limits of integration become:
- ρ: 0 to 1
- θ: 0 to 2π
- φ: 0 to π/2
Next, let's express the differential volume element in terms of spherical coordinates. The differential volume element in Cartesian coordinates is dx dy dz. In spherical coordinates, it becomes ρ^2 sin(φ) dρ dθ dφ.
Now, we can rewrite the integral in terms of spherical coordinates:
∫∫∫ [0,1]∫ [0,√(1-x^2 -y^2)]∫ 𝑧√(x^2 +y^2 +z^2) dz dy dx
becomes
∫∫∫ [0,π/2]∫ [0,2π]∫ [0,1] ρ^3 sin(φ) dρ dθ dφ √(ρ^2 +z^2)
To evaluate the integral, we compute each integral one by one, starting from the innermost integral:
∫ [0,π/2]∫ [0,2π]∫ [0,1] ρ^3 sin(φ) dρ dθ dφ √(ρ^2 +z^2)
Next, integrate with respect to ρ:
∫ [0,π/2]∫ [0,2π] [ρ^4/4]√(ρ^2 +z^2)| [0,1] dθ dφ
Evaluate the limits of ρ:
∫ [0,π/2]∫ [0,2π] [1/4]√(1 +z^2) - 0 dθ dφ
Simplify the integral:
∫ [0,π/2]∫ [0,2π] [1/4]√(1 +z^2) dθ dφ
Integrate with respect to θ:
∫ [0,π/2] [θ] [0,2π] [1/4]√(1 +z^2) dφ
Evaluate the limits of θ:
∫ [0,π/2] 2π [1/4]√(1 +z^2) dφ
Simplify the integral:
π [1/4]√(1 +z^2) [0,π/2]
Evaluate the limits of φ:
π/2 [1/4]√(1 +z^2)
Simplify the integral:
(π/2)[1/4]√(1 +z^2)
So, the equivalent integral in spherical coordinates is (π/2)[1/4]√(1 +z^2).