Find the volume of the indicated region.
-the region common to the interiors of the cylinders $x ^ { 2 } + y ^ { 2 } = 100$ and $x ^ { 2 } + z ^ { 2 } = 100$
A) $\frac { 4000 } { 3 }$
B) 4000
C) $\frac { 16000 } { 3 }$
D) $\frac { 2000 } { 3 }$

Question

Quizplus · Accepted Answer

The region common to the interiors of the two cylinders $x^2 + y^2 = 100$ and $x^2 + z^2 = 100$ forms a solid of revolution with a square cross-section when viewed along the axis of revolution. The side length of the square is $2\sqrt{100} = 20$, and the height of the solid is also 20 (the diameter of the circles formed by the cylinders). The volume of this solid can be calculated as the area of the square cross-section times the height, which is $20^2 \times 20 = 400 \times 20 = 8000$. However, since the question seems to be looking for the volume within both cylinders, which forms a three-dimensional "Steinmetz solid," the calculation involves integrating the area of overlap of the two cylinders along the axis. The correct formula for the volume of the intersecting region of two cylinders of equal radius $r$ is $V = 16r^3/3$. Substituting $r = 10$ (since $r^2 = 100$), we get $V = 16 \times 10^3 / 3 = 16000 / 3$.

Find the Volume of the Indicated Region x2+y2=100x ^ { 2 } + y ^ { 2 } = 100x2+y2=100

Find the Volume of the Indicated Region $x ^ { 2 } + y ^ { 2 } = 100$