Find the volume of the indicated region.
-the region enclosed by the cylinder $x ^ { 2 } + y ^ { 2 } = 49$ and the planes $z = 0$ and $x + y + z = 14$
A) $343 \pi$
B) $686 \pi$
C) $1029 \pi$
D) $1372 \pi$

Question

Quizplus · Accepted Answer

The cylinder $x^2 + y^2 = 49$ can be written as $z=0$ and $z=\sqrt{49-x^2-y^2}$. The volume between $z=0$ and $z=14-x-y$ is the integral of the function $\sqrt{49-x^2-y^2}$ over the region enclosed by the cylinder. Using cylindrical coordinates, the integral becomes 
$$\int_0^{2\pi}\int_0^7\int_0^{14-r\cos\theta}r\sqrt{49-r^2}dzdrd\theta=\frac{343}{3}\pi.$$ 
Therefore, the answer is B.

Find the Volume of the Indicated Region x2+y2=49x ^ { 2 } + y ^ { 2 } = 49x2+y2=49

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