Final answer:
The limits for the integral are from 0 to 2 for the radial direction, from -1 to 6 for the z direction, and from 0 to 2π for the angular direction. Therefore, F = 6
Step-by-step explanation:
The student is asking how to set up an iterated integral for the function f(x, y, z) = x² + y² + z² over the volume W of a solid cylinder centered about the z-axis, with a height of 7 and a base radius of 2, and with its base starting at z = -1. Since the cylinder is symmetric about the z-axis and has a circular base, it is natural to use cylindrical coordinates for the integral. To evaluate the integral ∫ f dV, we need to express f in cylindrical coordinates, so f(r, θ, z) = r² + z² where r is the radial distance, θ (theta) is the angle around the axis, and z is the height.
The limits for r will be from 0 to 2 (the radius of the base), for θ from 0 to 2π (completing a full circle around the z-axis), and for z from -1 to 6 (since the height of the cylinder is 7 and it starts at z = -1).
The iterated integral in cylindrical coordinates will be:
∫∫∫ f(r, θ, z) dz dr dθ with limits for z from E = -1 to F = 6, for r from C = 0 to D = 2, and for θ from A = 0 to B = 2π.