Find the area of the surface given by $z = f ( x , y ) \text { over the region } R \text {. }$ $\begin{array} { l }
f ( x , y ) = x y \
R = \left\{ ( x , y ) : x ^ { 2 } + y ^ { 2 } \leq 100 \right\}
\end{array}$
A) $\frac { 2 } { 3 } ( 101 \sqrt { 101 } - 1 ) \pi$
B) $\frac { 2 } { 3 } ( 1 - 101 \sqrt { 101 } ) \pi$
C) $\frac { 2 } { 3 } ( 101 \sqrt { 101 } - 1 )$
D) $\frac { 3 } { 4 } ( 101 \sqrt { 101 } - 1 ) \pi$
E) $\frac { 2 } { 3 } ( 100 \sqrt { 101 } - 1 ) \pi$

Question

Find the area of the surface given by $z = f ( x , y ) \text { over the region } R \text {. }$ $\begin{array} { l } 
f ( x , y ) = x y \
R = \left\{ ( x , y ) : x ^ { 2 } + y ^ { 2 } \leq 100 \right\}
\end{array}$
A) $\frac { 2 } { 3 } ( 101 \sqrt { 101 } - 1 ) \pi$
B) $\frac { 2 } { 3 } ( 1 - 101 \sqrt { 101 } ) \pi$
C) $\frac { 2 } { 3 } ( 101 \sqrt { 101 } - 1 )$
D) $\frac { 3 } { 4 } ( 101 \sqrt { 101 } - 1 ) \pi$
E) $\frac { 2 } { 3 } ( 100 \sqrt { 101 } - 1 ) \pi$

Quizplus · Accepted Answer

The area of the surface $z = f(x, y) = xy$ over the region $R$, where $R$ is defined by $x^2 + y^2 \leq 100$, can be calculated using the formula for the surface area of a graph of a function over a region in the xy-plane. The correct formula involves integrating the square root of $1 + (\frac{\partial f}{\partial x})^2 + (\frac{\partial f}{\partial y})^2$ over the region $R$. The derivatives of $f(x, y) = xy$ with respect to $x$ and $y$ are $y$ and $x$, respectively. Thus, the integrand becomes the square root of $1 + x^2 + y^2$, which, when integrated over the circular region defined by $x^2 + y^2 \leq 100$, leads to the correct answer after applying the appropriate integration techniques, including possibly converting to polar coordinates for easier computation.

Find the Area of the Surface Given By z=f(x,y) over the region R. z = f ( x , y ) \text { over the region } R \text {. }z=f(x,y) over the region R.

Find the Area of the Surface Given By $z = f ( x , y ) \text { over the region } R \text {. }$