 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2010/paper-2/18/a/solution

Split the integrand into its three coordinate contributions. Each integrates independently over the rectangular box:

I = (\frac{2}{3}) (4) (6) + (2) (\frac{16}{3}) (6) + (2) (4) (18) = 16 + 64 + 144 = 224 .

For example, the last factor is

\int_{- 3}^{3} z^{2} d z = 18

. No spherical or cylindrical Jacobian is needed for this Cartesian integral.

 New to topics? Read the docs here!