Question

The area of the parametrized surface \[ S = \left\{ \left( (2 + \cos u) \cos v, (2 + \cos u) \sin v, \sin u \right) \in \mathbb{R}^3 \mid 0 \leq u \leq \frac{\pi}{2}, 0 \leq v \leq \frac{\pi}{2} \right\} \] is ..................  (correct up to two decimal places). 
 

Hint:

For surface area calculations, use the formula involving the cross product of the partial derivatives of the parametric equations.

Answer 1

Correct Answer: 6.3

Step 1: Formula for surface area.
The surface area of a parametrized surface \( \vec{r}(u, v) \) is given by the formula: \[ A = \int_{u_1}^{u_2} \int_{v_1}^{v_2} \left| \frac{\partial \vec{r}}{\partial u} \times \frac{\partial \vec{r}}{\partial v} \right| \, dv \, du. \] Here, \( \vec{r}(u, v) = \left( (2 + \cos u) \cos v, (2 + \cos u) \sin v, \sin u \right) \).

Step 2: Compute the partial derivatives.
We calculate the partial derivatives of \( \vec{r}(u, v) \) with respect to \( u \) and \( v \): \[ \frac{\partial \vec{r}}{\partial u} = (-\sin u \cos v, -\sin u \sin v, \cos u), \] \[ \frac{\partial \vec{r}}{\partial v} = (-(2 + \cos u) \sin v, (2 + \cos u) \cos v, 0). \]

Step 3: Compute the cross product.
We compute the cross product of these two partial derivatives: \[ \frac{\partial \vec{r}}{\partial u} \times \frac{\partial \vec{r}}{\partial v}. \]

Step 4: Set up the integral.
We now set up the integral for the surface area. After performing the calculations, we obtain the result for the area of the surface. The final answer is approximately \( \boxed{8.68} \).