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Question

Write a double integral that represents the surface area of $z=f(x, y)$ over the region $R$. Use a computer algebra system to evaluate the double integral.

$f(x, y)=4-x^2-y^2$ $R=\{(x, y): 0 \leq x \leq 1,0 \leq y \leq 1\}$

Solution

VerifiedAnswered 1 year ago

Answered 1 year ago

Step 1

1 of 3Begin by finding the partial derivatives of the function. Since

$f_x(x,y)=-2x,\ f_y(x,y)=-2y,$

by applying the definition of surface area,

$\begin{aligned} S&=\iint\limits_R\sqrt{1+(-2x)^2+(-2y)^2}\ dA\\ &=\iint\limits_R\sqrt{1+4(x^2+y^2)}\ dA. \end{aligned}$

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