91Ó°ÊÓ

Calculate the partial derivative of \(x(u, v)\) with respect to \(u\) and \(v\). The partial derivatives are given as: $$ x_u(u, v) = \frac{\partial x}{\partial u} = \left(\begin{array}{c}2 \sin u \cos u \cos v \\ 2 \sin u \cos u \sin v \\ \cos^2 u - \sin^2 u\end{array}\right), $$ $$ x_v(u, v) = \frac{\partial x}{\partial v} = \left(\begin{array}{c} -\sin^2 u \sin v \\ \sin^2 u \cos v \\ 0\end{array}\right). $$

Calculate the cross product of \(x_u\) and \(x_v\). The cross product is given as: $$ x_u \times x_v = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 2 \sin u \cos u \cos v & 2 \sin u \cos u \sin v & \cos^2 u - \sin^2 u \\ -\sin^2 u \sin v & \sin^2 u \cos v & 0 \end{vmatrix}. $$ Solving the determinant, we have: $$ x_u \times x_v = \left(\begin{array}{c}-(\cos^2 u - \sin^2 u) \sin^2 u \cos v \\ -(\cos^2 u - \sin^2 u) \sin^2 u \sin v \\ 2 \sin^3 u \cos u \end{array}\right). $$

Calculate the magnitude of the cross product \(x_u \times x_v\). The magnitude is given as: $$ ||x_u \times x_v|| = \sqrt{(\text{-}(\cos^2 u - \sin^2 u) \sin^2 u \cos v)^2 + (\text{-}(\cos^2 u - \sin^2 u) \sin^2 u \sin v)^2 + (2 \sin^3 u \cos u)^2}. $$ Simplify the expression: $$ ||x_u \times x_v|| = \sin^3 u\sqrt{4 (\sin^2 u \cos^2 v + \sin^2 u \sin^2 v)}. $$ Further simplification gives: $$ ||x_u \times x_v|| = 2 \sin^3 u . $$

Integrate the magnitude of the cross product over the given range for \(u\) and \(v\) to find the surface area: $$ \text{Surface Area} = \iint_F ||x_u \times x_v|| \, du\, dv = \int_0^{\pi} \int_0^{2\pi} 2 \sin^3 u \, dv\, du. $$ Now evaluate the integral: $$ \text{Surface Area} = \int_0^{\pi} \left[ 4 \pi \sin^3 u \right] \, du = 4\pi \int_0^{\pi} \sin^3 u \, du. $$ Using the substitution \(t = \cos u\) and \(dt = -\sin u\, du\), we can rewrite the integral as: $$ \text{Surface Area} = 4\pi \left[-\int_1^{-1} t^2 dt\right] = 4\pi \left[\frac{1}{3}\left(t^3\right)\Big|_1^{-1}\right] = 4\pi \left[\frac{2}{3}\right] = \frac{8}{3} \pi. $$ The surface area of the surface with the given parameterization is \(\frac{8}{3} \pi\).