Find the surface area of the surface S.
-$S$ is the upper cap cut from the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 169$ by the cylinder $x ^ { 2 } + y ^ { 2 } = 25$.
A) $650 \pi$
B) $26 \pi$
C) $- 26 \pi$
D) 13

Question

Quizplus · Accepted Answer

The sphere has a radius of 13, and the cylinder has a radius of 5. The cap is the part of the sphere above the cylinder, which intersects the sphere at $z = \sqrt{169 - 25} = 12$. The surface area of the cap is given by $2 \pi r h$, where $r = 5$ and $h = 13 - 12 = 1$, resulting in $2 \pi \times 5 \times 1 = 10 \pi$. However, the correct calculation involves integrating over the circular region, leading to the surface area $26 \pi$.

Find the Surface Area of the Surface S SSS Is the Upper Cap Cut from the Sphere

Find the Surface Area of the Surface S $S$ Is the Upper Cap Cut from the Sphere