Calc 4 ### Surface Integral CalculationTo evaluate the surface integral \[ \iint_S \sqrt{9 + 3z^2} \, dS \]for the surface \( S \) defined by the parametric equations:\[ \mathbf{r}(u, v) = \left[ 6 \cos(u) \sin(v), 6 \sin(u) \sin(v), 3 \cos(v) \right] \]we must consider the given bounds for the parameters \( u \) and \( v \):\[ 0 \leq u \leq \frac{\pi}{2}, \quad 0 \leq v \leq \frac{\pi}{2}. \]### Explanation of Parameters and Surface- \( \mathbf{r}(u, v) \) specifies the parameterization of the surface \( S \).- \( u \) and \( v \) are the parameters that vary within the given intervals.- \( 6 \cos(u) \sin(v) \) and \( 6 \sin(u) \sin(v) \) represent the x and y coordinates respectively.- \( 3 \cos(v) \) represents the z coordinate. The aim is to integrate the given function \( \sqrt{9 + 3z^2} \) over the surface area defined by the parametric equations.### Important ConsiderationsFor this problem, understanding the parametric representation of the surface and being able to compute the surface element \( dS \) will be crucial for carrying out the surface integral. The function under the integral involves the z-component of the parametric equations, which needs to be accounted for during the integration process.

Name: Calc 4 ### Surface Integral CalculationTo evaluate the surface integra
Uploaded: 2024-03-20T07:06:41.000Z
Channel: Bartleby
Description: Calc 4 ### Surface Integral CalculationTo evaluate the surface integral \[ \iint_S \sqrt{9 + 3z^2} \, dS \]for the surface \( S \) defined by the parametric equations:\[ \mathbf{r}(u, v) = \left[ 6 \cos(u) \sin(v), 6 \sin(u) \sin(v), 3 \cos(v) \right