## Related questions with answers

Surface Integrals $\iint_S G(\mathbf{r}) d A$. Using $\iint_S G(\mathbf{r}) d A=\iint_R G(\mathbf{r}(u, v))|\mathbf{N}(u, v)| d u d v$ or $\iint_S G(\mathbf{r}) d A=\iint_{R^*} G(x, y, f(x, y)) \sqrt{1+\left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial f}{\partial y}\right)^2} d x d y$, evaluate thise integral for the given data. (Show the details.)

$G=(1+9 x z)^{3 / 2}, \quad S: \mathbf{r}=\left[\begin{array}{lll}u, & v, & u^3\end{array}\right], \quad 0 \leqq u \leqq 1,-2 \leqq v \leqq 2$

Solution

VerifiedWe know that

$\int\int_S G(\mathbf{r}) dA=\int\int_R G(\mathbf{r}(u,v))|\mathbf{N}(u,v)| du dv$

where

$\mathbf{N}(u,v)=\mathbf{r}_u\times \mathbf{r}_v$

By plugging $x=u$, $y=v$ and $z=u^3$ in the given function $G(x,y)$ we get

$\begin{aligned} \color{#4257b2} G(u,v)=(1+9u^4)^{3/2} \end{aligned}$

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