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

Short answer

The double integral of the given function that represents the surface area is \(A = \iint_R \sqrt{1 + 4x^{2} + 4y^{2}} dx dy\). Its exact value can be computed using a computer algebra system.

Step-by-step solution

Determine the Expression for Surface Area

The general formula to compute the surface area of \(z = f(x, y)\) using double integrals is \(A = \iint_R \sqrt{1 + (f'_x(x, y))^2 + (f'_y(x, y))^2} dx dy\). \(f'_x(x, y)\) and \(f'_y(x, y)\) denote the partial derivatives of \(f(x, y)\) with respect to \(x\) and \(y\) respectively. Compute these partial derivatives.

Evaluate the Partial Derivatives

The partial derivative of \(f(x, y)\) with respect to \(x\) is given by differentiating \(f(x, y)\) while keeping \(y\) constant. So \(f'_x(x, y) = -2x\). Same way, the partial derivative of \(f(x, y)\) with respect to \(y\) is \(f'_y(x, y) = -2y\).

Substitute Partial Derivatives into the Surface Area Formula

Substitute these partial derivatives into the surface area formula, so the expression becomes \(A = \iint_R \sqrt{1 + (-2x)^2 + (-2y)^2} dx dy\). This simplifies to \(A = \iint_R \sqrt{1 + 4x^{2} + 4y^{2}} dx dy\).

Compute the Double Integral

Substitute the given limits for \(x\) and \(y\) into the double integral and compute. This can be done manually or using a computer algebra system, such as Mathematica or Wolfram Alpha.