Find the volume of the indicated region.
-the region bounded by the paraboloid $z = 49 - x ^ { 2 } - y ^ { 2 }$ and the $x y$-plane
A) $\frac { 2401 } { 2 } \pi$
B) $\frac { 343 } { 2 } \pi$
C) $\frac { 2401 } { 3 } \pi$
D) $\frac { 343 } { 3 } \pi$

Question

Quizplus · Accepted Answer

The region is a circular disk of radius 7, so the volume is$$V = \int_{-7}^7 \int_{-\sqrt{49-x^2}}^{\sqrt{49-x^2}} (49 - x^2 - y^2) \ dy \ dx$$$$= \int_{-7}^7 \left[ 49y - \frac{y^3}{3} - xy^2 \right]_{-\sqrt{49-x^2}}^{\sqrt{49-x^2}} \ dx$$$$= \int_{-7}^7 \left( 98\sqrt{49-x^2} - \frac{2}{3}(49-x^2)^{3/2} \right) \ dx$$$$= \left[ 98\left(\frac{1}{2}\right)(49x - \frac{x^3}{3}) - \frac{2}{3}\left(\frac{2}{5}\right)(49-x^2)^{5/2} \right]_{-7}^7$$$$= \frac{2401}{2}\pi$$

Find the Volume of the Indicated Region z=49−x2−y2z = 49 - x ^ { 2 } - y ^ { 2 }z=49−x2−y2

Find the Volume of the Indicated Region $z = 49 - x ^ { 2 } - y ^ { 2 }$