Verified answer

Integrate[Integrate r^3 * sin theta * r dr From r = 0 to r = 5 + 2cos(theta)] dtheta From theta = 0 to theta = pi/2

sin theta * (1/5)r^5 Evaluate r = 0 to r = 5 + 2cos(theta)

Integrate [(1/5) sin theta * (5 + 2cos(theta))^5] dtheta From theta = 0 to theta = pi/2

u = 5 + 2cos(theta), du = -2 sin theta dtheta

(-1/10) du = (1/5)sin theta dtheta

theta = 0, 5 + 2cos(theta) = 7

theta = pi/2, 5 + 2cos(theta) = 5

Integrate (-1/10) u^5 du From u = 7 to u = 5

(1/10) * Integrate u^5 du From u = 5 to u = 7

(1/60) * [u^6 From u = 5 to u = 7]

(1/60) * [7^6 - 5^6] = 8502/5 = 1700.4