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 $\iint _ { S } f ( x , y , z ) d S$ can be evaluated by projecting the surface $S$ onto a coordinate plane and then integrating. The given surface is $2x + 4y + 3z = 24$, which intersects the coordinate planes at $x = 12$, $y = 6$, and $z = 8$ in the first octant, forming a triangle on the plane. The function to be integrated over this surface is $f(x, y, z) = x + y$. However, we need to express $z$ in terms of $x$ and $y$ to use it in the surface integral formula. From the equation of the plane, we have $z = \frac{24 - 2x - 4y}{3}$.The area element $dS$ on the plane can be expressed as $dS = \sqrt{1 + \left(\frac{\partial z}{\partial x}\right)^2 + \left(\frac{\partial z}{\partial y}\right)^2}dA$, where $dA$ is the differential area element in the $xy$-plane. The partial derivatives of $z$ with respect to $x$ and $y$ are $\frac{\partial z}{\partial x} = -\frac{2}{3}$ and $\frac{\partial z}{\partial y} = -\frac{4}{3}$, respectively. Substituting these into the expression for $dS$ gives $dS = \sqrt{1 + \left(-\frac{2}{3}\right)^2 + \left(-\frac{4}{3}\right)^2}dA = \sqrt{1 + \frac{4}{9} + \frac{16}{9}}dA = \sqrt{\frac{29}{9}}dA = \frac{\sqrt{29}}{3}dA$.To find the integral over the surface, we integrate $f(x, y, z)$ over the projected triangle in the $xy$-plane, using $dA = dx dy$ and the limits from $0$ to $12$ for $x$ and from $0$ to $6 - \frac{1}{2}x$ for $y$, considering the line $2x + 4y = 24$ or $y = 6 - \frac{1}{2}x$. The integral becomes:$$\iint_{S} (x + y) dS = \int_{0}^{12} \int_{0}^{6 - \frac{1}{2}x} (x + y) \frac{\sqrt{29}}{3} dy dx$$Evaluating this double integral gives the area of the projected triangle times $\frac{\sqrt{29}}{3}$, which results in $72\sqrt{29}$ after performing the integration and considering the geometry of the problem.

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

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