Question

Use double integration to find the volume of the solid bounded by the paraboloid z= 9x^2 + y^2, below by the plane z = 0, and laterally by the planes x = 0, y = 0, x = 3, and y = 2.

Answer 1

$$V=\displaystyle\int_{0}^3\displaystyle\int_{0}^2(9x^2+y^2)dydx$$

$$=\displaystyle\int_{0}^3[9x^2y+y^3/3]\begin{matrix} 2\\ 0 \end{matrix}dx$$ $$=\displaystyle\int_{0}^3(18x^2+8/3)dx$$

$$=[6x^3+8x/3]\begin{matrix} 3 \\ 0 \end{matrix}$$

$$=170 (cubic\ units)$$