Which of the following could represent the triple integral in cylindrical coordinates where Q is the region bounded below by and above by ?

Question

Which of the following could represent the triple integral  in cylindrical coordinates where Q is the region bounded below by  and above by  ?

Quizplus · Accepted Answer

In cylindrical coordinates, the triple integral can be expressed as:
∫∫∫Q f(r,θ,z) r dz dr dθ

Since Q is bounded above and below by two equations involving z, we can express the limits of integration as follows:
z limits: from z= r^2 to z= 4r
r limits: from 0 to 2
θ limits: from 0 to 2π

Therefore, the integral can be expressed as:
∫ from 0 to 2π ∫ from 0 to 2 ∫ from r^2 to 4r f(r,θ,z) r dz dr dθ

Looking at the answer choices, we can see that only choice C contains the values for r and z that are present in the limits of integration. Therefore, choice C represents the triple integral in cylindrical coordinates.

Which of the Following Could Represent the Triple Integral