Find the volume of the indicated region.
-the region bounded by the cylinder $x ^ { 2 } + y ^ { 2 } = 49$ and the planes $z = 0$ and $x + z = 8$
A) $392 \pi$
B) $448 \pi$
C) $56 \pi$
D) $3136 \pi$

Question

Quizplus · Accepted Answer

The volume of the region can be found by integrating the area of circular cross-sections along the z-axis. The cylinder $x^2 + y^2 = 49$ has a radius of 7, so its cross-sectional area is $\pi(7)^2 = 49\pi$. The height of the cylinder, determined by the plane $x + z = 8$ (since $x$ can be at most 7, $z$ ranges from 0 to 1 when $x = 7$), is 8. Therefore, the volume is the area of the base times the height, $49\pi \times 8 = 392\pi$.

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

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