Three Surface Integrals

Question

Could someone assist with the following three surface integrals?

Q1 The portion of the cone $z=\sqrt{x^2+y^2}$ that lies inside the cylinder $x^2+y^2 =2x$.

Q2 The portion of the paraboloid $z=1-x^2-y^2$ that lies above the $xy$-plane.

Q3 The portion of the paraboloid $2z = x^2+y^2$ that is inside the cylinder $x^2+y^2=8$.

Any assistance will be greatly appreciated.

Answer 1

You have $z=f(x,y)$ and the surface area is $$\int\int_A\sqrt{1+f_x^2+f_y^2}dxdy$$ where $A$ is the projection of the surface area on the $xy$-plane.
Use polar coordinates and so $dA=rdrd\theta$

1. $r=0\to r=2\cos\theta,\theta=-\pi/2\to\pi/2$
2. $r=0\to r=1,\theta=0\to2\pi$
3. $r=0\to r=\sqrt8,\theta=0\to2\pi$
   For the first one, $z=\sqrt{x^2+y^2}$ and so $$\int\int_A\sqrt{1+\frac{x^2}{x^2+y^2}+\frac{y^2}{x^2+y^2}}dxdy \\=\int_{-\pi/2}^{\pi/2}\int_0^{2\cos \theta}\sqrt{2}rdrd\theta \\=\frac{\sqrt{2}}{2}\int_{-\pi/2}^{\pi/2}r^2|_0^{2\cos \theta}d\theta$$

Answer 2

$\newcommand{\+}{^{\dagger}}% \newcommand{\angles}[1]{\left\langle #1 \right\rangle}% \newcommand{\braces}[1]{\left\lbrace #1 \right\rbrace}% \newcommand{\bracks}[1]{\left\lbrack #1 \right\rbrack}% \newcommand{\ceil}[1]{\,\left\lceil #1 \right\rceil\,}% \newcommand{\dd}{{\rm d}}% \newcommand{\down}{\downarrow}% \newcommand{\ds}[1]{\displaystyle{#1}}% \newcommand{\equalby}[1]{{#1 \atop {= \atop \vphantom{\huge A}}}}% \newcommand{\expo}[1]{\,{\rm e}^{#1}\,}% \newcommand{\fermi}{\,{\rm f}}% \newcommand{\floor}[1]{\,\left\lfloor #1 \right\rfloor\,}% \newcommand{\half}{{1 \over 2}}% \newcommand{\ic}{{\rm i}}% \newcommand{\iff}{\Longleftrightarrow} \newcommand{\imp}{\Longrightarrow}% \newcommand{\isdiv}{\,\left.\right\vert\,}% \newcommand{\ket}[1]{\left\vert #1\right\rangle}% \newcommand{\ol}[1]{\overline{#1}}% \newcommand{\pars}[1]{\left( #1 \right)}% \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\pp}{{\cal P}}% \newcommand{\root}[2][]{\,\sqrt[#1]{\,#2\,}\,}% \newcommand{\sech}{\,{\rm sech}}% \newcommand{\sgn}{\,{\rm sgn}}% \newcommand{\totald}[3][]{\frac{{\rm d}^{#1} #2}{{\rm d} #3^{#1}}} \newcommand{\ul}[1]{\underline{#1}}% \newcommand{\verts}[1]{\left\vert\, #1 \,\right\vert}$ $\ds{\color{#f00}{\Huge\tt Q_{1}}}$: With $\vec{r} = x\,\hat{x} + y\,\hat{y} + \root{x^{2} + y^{2}}\,\hat{z}$ \begin{align} \color{#00f}{\large{\cal A}_{Q_{1}}}&= \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \verts{\partiald{\vec{r}}{x}\times\partiald{\vec{r}}{y}}\, \Theta\pars{2x - x^{2} - y^{2}}\,\dd x\,\dd y \\[3mm]&= \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \verts{\pars{\hat{x} + {x \over \root{x^{2} + y^{2}}}\hat{z}}\times \pars{\hat{y} + {y \over \root{x^{2} + y^{2}}}\hat{z}}}\, \Theta\pars{2x - x^{2} - y^{2}}\,\dd x\,\dd y \\[3mm]&= \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \verts{\hat{z} - {y\,\hat{y} \over \root{x^{2} + y^{2}}} + {x\,\hat{x} \over \root{x^{2} + y^{2}}}}\, \Theta\pars{2x - x^{2} - y^{2}}\,\dd x\,\dd y \\[3mm]&= \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \root{{x^{2} \over x^{2} + y^{2}} + {y^{2} \over x^{2} + y^{2}} + 1} \Theta\pars{2x - x^{2} - y^{2}}\,\dd x\,\dd y \\[3mm]&=\root{2}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \Theta\pars{2x - x^{2} - y^{2}}\,\dd x\,\dd y =\root{2}\int_{0}^{\infty}\dd r\,r \int_{0}^{2\pi}\dd\theta\,\Theta\pars{2r\cos\pars{\theta} - r^{2}} \\[3mm]&=\root{2}\int_{0}^{2\pi}\dd\theta \int_{0}^{\infty}\Theta\pars{2\cos\pars{\theta} - r}r\,\dd r =\root{2}\int_{0}^{2\pi}\dd\theta\,\Theta\pars{\cos\pars{\theta}} \int_{0}^{2\cos\pars{\theta}}r\,\dd r \\[3mm]&=2\root{2}\int_{0}^{2\pi}\dd\theta\,\Theta\pars{\cos\pars{\theta}} \cos^{2}\pars{\theta} = 2\root{2}\bracks{% \int_{0}^{\pi/2}\cos^{2}\pars{\theta}\,\dd\theta + \int_{3\pi/2}^{2\pi}\cos^{2}\pars{\theta}\,\dd\theta} \\[3mm]&= 2\root{2}\bracks{% \int_{0}^{\pi/2}\cos^{2}\pars{\theta}\,\dd\theta + \int_{0}^{\pi/2}\sin^{2}\pars{\theta}\,\dd\theta} =\color{#00f}{\large\root{2}\pi} \end{align}

$\ds{\color{#f00}{\Huge\tt Q_{2}}}$: Similarly, with $\vec{r} = x\,\hat{x} + y\,\hat{y} + \pars{1 - x^{2} - y^{2}}\hat{z}$: $$ \verts{\partiald{\vec{r}}{x}\times\partiald{\vec{r}}{y}} =\verts{\pars{\hat{x} - 2x\,\hat{z}}\times\pars{\hat{y} - 2y\,\hat{z}}} =\verts{\hat{z} + 2y\,\hat{y} - 2x\,\hat{x}} =\root{4\pars{x^{2} + y^{2}} + 1} $$ \begin{align} \color{#00f}{\large{\cal A}_{Q_{2}}}&=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \root{4\pars{x^{2} + y^{2}} + 1}\Theta\pars{1 - x^{2} - y^{2}}\,\dd x\,\dd y \\[3mm]&=\int_{0}^{\infty}\dd r\,r\int_{0}^{2\pi}\dd\theta\,\root{4r^{2} + 1} \Theta\pars{1 - r^{2}} =2\pi\int_{0}^{1}\root{4r^{2} + 1}r\,\dd r \\[3mm]&=\pi\int_{0}^{1}\root{4r + 1}\,\dd r =\left.{1 \over 6}\,\pars{4r + 1}^{3/2}\right\vert_{0}^{1} =\color{#00f}{\large{1 \over 6}\pars{5\root{5} - 1}\pi} \end{align}

$\ds{\color{#f00}{\Huge\tt Q_{3}}}$: $\ds{\vec{r} = x\,\hat{x} + y\,\hat{y} + \half\,\pars{x^{2} + y^{2}}\hat{z}}$ $$ \verts{\partiald{\vec{r}}{x}\times\partiald{\vec{r}}{y}} =\verts{\pars{\hat{x} + x\,\hat{z}}\times\pars{\hat{y} + y\,\hat{z}}} =\verts{\hat{z} - y\,\hat{y} + x\,\hat{x}} = \root{x^{2} + y^{2} + 1} $$ \begin{align} \color{#00f}{\large{\cal A}_{Q_{3}}}&= \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\root{x^{2} + y^{2} + 1} \Theta\pars{8 - x^{2} - y^{2}}\,\dd x\,\dd y =2\pi\int_{0}^{2\root{2}}\root{r^{2} + 1}r\,\dd r \\[3mm]&=\pi\int_{0}^{8}\root{r + 1}\,\dd r =\pi\left.{2 \over 3}\,\pars{r + 1}^{3/2}\right\vert_{0}^{8} ={2 \over 3}\,\pi\pars{9^{3/2} - 1} =\color{#00f}{\large{52 \over 3}\,\pi} \end{align}