Find the volume of the indicated region.
-the region bounded by the coordinate planes, the parabolic cylinder $z = 9 - x ^ { 2 }$, and the plane $y = 9$
A) 162
B) 486
C) $\frac { 729 } { 4 }$
D) $\frac { 729 } { 2 }$

Question

Quizplus · Accepted Answer

The volume of the region can be found by integrating the function $z = 9 - x^2$ over the appropriate domain. The region is bounded by $x^2$ in the $x$-direction from $x = -3$ to $x = 3$ (since $9 - x^2 = 0$ gives $x = \pm3$), and by $y = 9$ in the $y$-direction. The volume $V$ is given by the integral $\int_{-3}^{3} \int_{0}^{9} (9 - x^2) dy dx$. Integrating, we find $V = \int_{-3}^{3} 9y - yx^2 |_{0}^{9} dx = \int_{-3}^{3} 81 - 9x^2 dx = 162x - 3x^3 |_{-3}^{3} = 162$.

Find the Volume of the Indicated Region z=9−x2z = 9 - x ^ { 2 }z=9−x2

Find the Volume of the Indicated Region $z = 9 - x ^ { 2 }$