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Volume, area & line integrals

Multivariate Calculus
Grigorios Kostakos
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Volume, area & line integrals

Let $E$ be the surface with parametric representation
\begin{align*}
\end{align*} and $S_{+}$ the upper hemisphere of the sphere $x^2+y^2+z^2=36$.
1. Find the volume of the solid $\Sigma$ which is enclosed by the surfaces $E,\, S_{+}$ and the plane $z=0$.
2. Find the area of the surface $(S_{+})\cap \Sigma$.
3. Let the function $f:{\mathbb{R}}^3\longrightarrow{\mathbb{R}}\,; \; f(x,y,z)=x\,z\,(36-z^2)^2$ and $c$ the curve which is the section of the surface $\partial\Sigma$ and the half-plane $\Pi: \big\{(x,0,z)\in{\mathbb{R}}^3\;|\; x\geqslant0 \big\}$. Find the line integral $\oint_{c}f\,ds$.
4. Let the vector field $\overline{F}:{\mathbb{R}}^3\longrightarrow{\mathbb{R}}^3\,;\quad\overline{F}(x,y,z)=\left({y-x\,,\,y^3z\,,\,z^2}\right)\,.$ Find the line integral $\oint_{c}\overline{F}\cdot d\overline{r}$.