Find the area of the surface given by $f ( x , y ) = 5 - x ^ { 2 }$
$R$ : rectangle with vertices $( 0,0 ) , ( 5,0 ) , ( 5,5 ) , ( 0,5 )$
A) $\frac { 5 } { 4 } ( 10 \sqrt { 101 } - \ln ( 10 + \sqrt { 101 } ) )$
B) $\frac { 5 } { 4 } ( 10 \sqrt { 101 } + \ln ( 10 - \sqrt { 101 } ) )$
C) $5 \left( 5 \sqrt { 101 } + \frac { 1 } { 2 } \ln ( 5 + \sqrt { 101 } ) \right)$
D) $5 \left( 5 \sqrt { 101 } - \frac { 1 } { 2 } \ln ( 5 + \sqrt { 101 } ) \right)$
E) $\frac { 5 } { 4 } ( 10 \sqrt { 101 } + \ln ( 10 + \sqrt { 101 } ) )$

Question

Quizplus · Accepted Answer

The area of a surface $z = f(x, y)$ over a region $R$ in the $xy$-plane is given by the double integral $\iint_R \sqrt{1 + \left(\frac{\partial f}{\partial x}\right)^2 + \left(\frac{\partial f}{\partial y}\right)^2} \,dx\,dy$. For $f(x, y) = 5 - x^2$, $\frac{\partial f}{\partial x} = -2x$ and $\frac{\partial f}{\partial y} = 0$. Substituting these into the formula and integrating over the given rectangle yields the correct answer. Choice E correctly calculates this integral.

Find the Area of the Surface Given By f(x,y)=5−x2f ( x , y ) = 5 - x ^ { 2 }f(x,y)=5−x2

Find the Area of the Surface Given By $f ( x , y ) = 5 - x ^ { 2 }$