Convert xz dz dy dx to an integral in cylindrical coordinate

Convert xz dz dy dx to an integral in cylindrical coordinates of the form g(r, theta, z) dz dr d theta. Do NOT evaluate the integral.

Solution

x = rsin(theta) ---> dx = rcos(theta)d(theta) + sin(theta)dr y = rcos(theta) ---> dy = -rsin(theta)d(theta) + cos(theta)dr we have sqrt(x^2+y^2) = sqrt(r^2) = r and sqrt(16-x^2) = sqrt(16-r^2sin^(theta)) Substitute in the original equation and you have the required form.