Question

Use a computer algebra system to approximate the double integral that gives the surface area of the graph of \(f\) over the region \(R=\\{(x, y): 0 \leq x \leq 1,0 \leq y \leq 1\\}\) $$ f(x, y)=\frac{2}{5} y^{5 / 2} $$

Short answer

The surface area can be approximated by evaluating the given double integral, which after some simplifications can be calculated using a computer algebra system.

Step-by-step solution

Applying formula

To calculate the surface area, the following formula is used: \(A=\iint_{R}\sqrt{1+(\frac{\partial f}{\partial x})^2+(\frac{\partial f}{\partial y})^2} \,dx\,dy\). The partial derivatives need to be found.

Compute partial derivatives

Since the function does not depend on \(x\), its partial derivative is \(0\). The partial derivative with respect to \(y\) is \(\frac{\partial f}{\partial y}=\frac{1}{2}y^{3/2}\).

Calculate surface area

Substituting the partial derivatives into the formula for the area gives \(A=\iint_{R}\sqrt{1+0^2+(\frac{1}{2}y^{3/2})^2} \,dx\,dy\). Simplify the expression and solve the double integral with the given boundaries \(0 \leq x \leq 1\) and \(0 \leq y \leq 1\).

Evaluating the integral

The integral becomes \(A=\iint_{R}\sqrt{1+\frac{1}{4}y^3} \,dx\,dy\). Evaluate it by using a computer algebra system.