Chapter 15 - Multiple Integrals - 15.6 Triple Integrals - 15.6 Exercises - Page 1077: 9

$\dfrac{27}{2}$

Let us consider $I=\iiint_E y dV$ Here, $ I=\int_{0}^3 \int_{0}^{x} \int_{x-y}^{x+y} dz dy dx= \int_{0}^3 \int_{0}^{x} [yz]_{x-y}^{x+y} dy dx$ Further, $\int_{0}^1 \int_{0}^{1} [xy(e^{2-x^2-y^2})-xy(e^0)] dy dx=\int_{0}^{3} yx+y^2-yx+y^2 dy dx$ and $ \int_0^3[\dfrac{2}{3} (y^3)]_0^x=[2x^4]_0^3$ Hence, $\iiint_E y dV=\dfrac{27}{2}$