Question

Area,find the area of the surface over the given region. Use a computer algebra system to verify your results. The surface of revolution \(\mathbf{r}(u, v)=\sqrt{u} \cos v \mathbf{i}+\sqrt{u} \sin v \mathbf{j}+\) \(u \mathbf{k},\) where \(0 \leq u \leq 4\) and \(0 \leq v \leq 2 \pi\)

Short answer

The surface area S of the surface of revolution can be calculated by setting up and evaluating the double integral as shown in the steps. The actual numerical value can be obtained by using a computer algebra system to evaluate the integral.

Step-by-step solution

Write down the formula for the surface area

The surface area S of a parametric surface defined by \(\mathbf{r}(u, v)\) is given by the double integral:\[ S = \int\int_D |\mathbf{r}_u \times \mathbf{r}_v| du dv \]where \(\mathbf{r}_u\) and \(\mathbf{r}_v\) are partial derivatives of \(\mathbf{r}\) with respect to u and v, and |.| denotes the magnitude (length) of the cross product vector.

Calculate \(\mathbf{r}_u\) and \(\mathbf{r}_v\)

The partial derivatives of \(\mathbf{r}(u, v)\) with respect to u and v are:\[\mathbf{r}_u = \frac{1}{2\sqrt{u}} \cos v \mathbf{i} + \frac{1}{2\sqrt{u}} \sin v \mathbf{j} + \mathbf{k} \]and\[\mathbf{r}_v = -\sqrt{u} \sin v \mathbf{i} + \sqrt{u} \cos v \mathbf{j}.\]

Compute the cross product \(\mathbf{r}_u \times \mathbf{r}_v\)

The cross product of \(\mathbf{r}_u\) and \(\mathbf{r}_v\) is:\[\mathbf{r}_u \times \mathbf{r}_v = \left(-\sqrt{u} \cos v, -\sqrt{u} \sin v, \frac{1}{2\sqrt{u}} \right).\]

Calculate the magnitude \(|\mathbf{r}_u \times \mathbf{r}_v|\)

The magnitude of the cross product vector is obtained from the Pythagorean theorem:\[|\mathbf{r}_u \times \mathbf{r}_v| = \sqrt{(-\sqrt{u} \cos v)^2 + (-\sqrt{u} \sin v)^2 + \left(\frac{1}{2\sqrt{u}} \right)^2} = \sqrt{u + \frac{1}{4}}.\]

Set up the double integral for the surface area

The surface area S of the surface is given by the double integral:\[S = \int_0^4 \int_0^{2\pi} \sqrt{u + \frac{1}{4}} du dv.\]

Calculate the surface area

Use a computer algebra system to calculate the double integral. Plug in the limits of integration and simplify until you get a numerical value for the surface area.