Evaluate $\iint _ { S } f ( x , y , z ) d S$
$f ( x , y , z ) = x + y ; S$ is the part of the plane $2 x + 4 y + 3 z = 24$ in the first octant.
A) $72 \sqrt { 29 }$
B) 0
C) $\sqrt { 403 }$
D) $\sqrt { 273 }$

Question

Quizplus · Accepted Answer

The surface integral over the plane in the first octant can be calculated using the formula for the surface integral of a function over a plane. The plane intersects the axes at (12,0,0), (0,6,0), and (0,0,8). The area of the triangle formed by these points is $\frac{1}{2} \times 12 \times 6 = 36$. The normal vector to the plane is (2,4,3), and its magnitude is $\sqrt{29}$. The integral of $f(x,y,z) = x + y$ over the surface is $36 \times \sqrt{29}$, which simplifies to $72 \sqrt{29}$.

Evaluate ∬Sf(x,y,z)dS\iint _ { S } f ( x , y , z ) d S∬S​f(x,y,z)dS

Evaluate $\iint _ { S } f ( x , y , z ) d S$