Find the surface area of the surface S.
-$S$ is the intersection of the plane $3 x + 4 y + 12 z = 7$ and the cylinder with sides $y = 4 x ^ { 2 }$ and $y = 8 - 4 x ^ { 2 }$.
A) $\frac { 13 } { 18 }$
B) $\frac { 104 } { 3 }$
C) $\frac { 13 } { 9 }$
D) $\frac { 104 } { 9 }$

Question

Quizplus · Accepted Answer

The surface area of the surface $S$ can be found by integrating over the region defined by the intersection of the given plane and cylinder. The cylinder is bounded by $y = 4x^2$ and $y = 8 - 4x^2$, and the plane is $3x + 4y + 12z = 7$. The surface area calculation involves projecting the surface onto a plane and integrating over the projected area, taking into account the slope of the surface to find the differential area element. The correct calculation leads to the surface area being $\frac{104}{9}$, which corresponds to choice D.

Find the Surface Area of the Surface S SSS Is the Intersection of the Plane

Find the Surface Area of the Surface S $S$ Is the Intersection of the Plane