Question

Set up but do not evaluate the integral over the region E inside the sphere S, and between the cones 4θ= 4πand 2θ=2π. Sketch the region of integration as part of your solution.

a) Triple integral
b) Double integral
c) Line integral
d) Surface integral

Answer 1

It is the appropriate choice due to the nature of the region of integration defined by the constraints within the problem Therefore,the correct Option is Option b) Double integral

Step-by-step explanation:

The region described lies within a sphere (S) and between two cones defined by the equations 4θ = 4π and 2θ = 2π. This indicates a double integral setup due to the constraints involving θ. Integrating over a region confined by cones and a sphere typically requires setting up a double integral in spherical coordinates, incorporating the bounds for θ, ϕ, and ρ.

The given information doesn't require evaluation, but understanding the boundaries—where θ ranges from π/2 to π and ϕ varies from 0 to π/4—helps establish the setup. The integral would involve the sphere's equation ρ = constant within the specified limits of θ and ϕ. This approach allows for calculating the volume within the constraints outlined by the cones and sphere.

Therefore,the correct Option is Option b) Double integral