Let N = \int \int \int_B xyz^2\ dV , where B is the cuboid bounded by the regions 0 \leq x \leq 1, -1 \leq y \leq 2 and 0 \leq z \leq 3. If N = \frac{a}{b}, where a and b are coprime positive integers. What is the value of a+b?

Answer 1

seems pretty straightforward:

v = ∫[0,1]∫[-1,2]∫[0,3] xyz^2 dz dy dx
= ∫[0,1]∫[-1,2] 1/3 xyz^3[0,3] dy dx
= 9∫[0,1]∫[-1,2] xy dy dx
= 9∫[0,1] 1/2 xy^2 [-1,2] dx
= 27/2 ∫[0,1] x dx
= 27/2 (1/2 x^2)[0,1]
= 27/4