Surface s is a part of the paraboloid z=4-x^2-y^2 that lies above the plane z=0.(6+7+7=20pt) a)Find the parametric equation vecr(u,v) of the surface with polar coordinates x=ucos(v), y=usin(v) and find the domain D for u and v.

OlmekinjP 2021-09-12 Answered

Surface s is a part of the paraboloid \(\displaystyle{z}={4}-{x}^{{2}}-{y}^{{2}}\) that lies above the plane \(z=0\).\((6+7+7=20pt)\)

a) Find the parametric equation \(\displaystyle\vec{{r}}{\left({u},{v}\right)}\) of the surface with polar coordinates \(\displaystyle{x}={u}{\cos{{\left({v}\right)}}},{y}={u}{\sin{{\left({v}\right)}}}\) and find the domain D for u and v.

b) Find \(\displaystyle\vec{{r}}_{{u}},\vec{{r}}_{{v}},\) and \(\displaystyle\vec{{r}}_{{u}}\cdot\vec{{r}}_{{v}}\).

c) Find the area of the surface

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Plainmath recommends

  • Ask your own question for free.
  • Get a detailed answer even on the hardest topics.
  • Ask an expert for a step-by-step guidance to learn to do it yourself.
Ask Question

Expert Answer

Viktor Wiley
Answered 2021-09-13 Author has 8343 answers
Given : \(\displaystyle{z}={4}-{x}^{{2}}-{y}^{{2}}\), \(\displaystyle{z}={0}\)
a) \(\displaystyle{x}={u}{\cos{{v}}},{y}={u}{\sin{{v}}},{z}={4}-{u}^{{2}}{{\cos}^{{2}}{v}}-{u}^{{2}}{\sin{{v}}}\)
\(\displaystyle{z}={4}-{u}^{{2}}\)
\(\displaystyle{r}{\left({u},{v}\right)}={<}{u}{\cos{{v}}},{u}{\sin{{v}}},{4}-{u}^{{2}}{>}\)
The domain of \(\displaystyle{0}\le{u}\le{2}\)
\(\displaystyle{0}\le{v}\le{2}\pi\)
b) \(\displaystyle{r}_{{u}}={<}{\cos{{v}}},{\sin{{v}}},-{2}{u}{>}\)
\(\displaystyle{r}_{{v}}={<}-{u}{\sin{{v}}},{u}{\cos{{v}}},{0}{>}\)
\(\displaystyle{r}{u}{x}{r}{v}={\left[\begin{array}{ccc} {i}&{j}&{k}\\{\cos{{v}}}&{\sin{{v}}}&-{2}{u}\\-{u}{\sin{{v}}}&{u}{\cos{{v}}}&{0}\end{array}\right]}\)
\(\displaystyle={i}{\left({0}+{2}{u}^{{2}}{\cos{{v}}}\right)}-{j}{\left(-{2}{u}^{{2}}{\sin{{v}}}\right)}+{k}{\left({u}{{\cos}^{{2}}{v}}+{u}{{\sin}^{{2}}{v}}\right)}={<}{2}{u}^{{2}}{\cos{{v}}},{2}{u}^{{2}}{\sin{{v}}},{u}{>}\)
c) Surface Area =\(\displaystyle{\int_{{0}}^{{2}}}\pi{\int_{{0}}^{{2}}}{\left|{r}_{{u}}{x}{r}_{{v}}\right|}{d}{u}{d}{v}\)
\(\displaystyle{\left|{r}_{{u}}{x}{r}_{{v}}\right|}=\sqrt{{4}}{u}^{{2}}{{\cos}^{{2}}{v}}+{4}{u}^{{4}}{{\sin}^{{2}}{v}}+{u}^{{2}}=\sqrt{{8}}{u}^{{4}}+{u}^{{2}}\)
\(\displaystyle={u}\sqrt{{1}}+{8}{u}^{{2}}\)
A=\(\displaystyle{\int_{{0}}^{{2}}}\pi{\int_{{0}}^{{2}}}{u}\sqrt{{1}}+{8}{u}^{{2}}{d}{u}{d}{v}\)
\(\displaystyle{1}+{84}^{{2}}={f}\)
\(\displaystyle={\int_{{0}}^{{2}}}\pi{d}{v}\int\sqrt{{f}}{d}\frac{{f}}{{16}}\)
\(\displaystyle{16}{u}{d}{u}={d}{f}\)
\(\displaystyle=\frac{{1}}{{16}}\int{v}{\int_{{0}}^{{2}}}\pi{\left[\frac{{f}^{{\frac{{3}}{{2}}}}}{{\frac{{3}}{{2}}}}\right]}_{{{u}={0}}}\)
\(\displaystyle=\frac{\pi}{{8}}\cdot\frac{{2}}{{3}}{\left[\frac{{\left({1}+{8}{u}^{{2}}\right)}^{{3}}}{{2}}{\int_{{0}}^{{2}}}\right.}\)
\(\displaystyle=\frac{\pi}{{12}}{\left[\frac{{\left({33}\right)}^{{3}}}{{2}}-{1}\right]}_{{u}}\)
Have a similar question?
Ask An Expert
31
 

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Relevant Questions

asked 2021-09-07
Find the area of the surface.
The part of the paraboloid
\(\displaystyle{z}={1}−{x}^{{2}}−{y}^{{2}}\)
that lies above the plane
\(\displaystyle{z}=−{6}\)
asked 2021-09-14

Find parametric equations for x, y, and z in terms of the polar coordinates r and \(\displaystyle\theta\) to determine the points on the portion of the paraboloid \(x + y + z = 5\) that is on or above the plane \(z=4\)

asked 2021-09-02

1) Find the area of the part of the plane
\(4x + 3y + z = 12\)
that lies in the first octant.
2) Use polar coordinates to find the volume of the given solid.
Bounded by the paraboloid \(\displaystyle{z}={5}+{2}{x}^{{2}}+{2}{y}^{{2}}\) and the plane z = 11 in the first octant

asked 2021-06-08

1) Find the area of the part of the plane
\(4x + 3y + z = 12\)
that lies in the first octant.
2) Use polar coordinates to find the volume of the given solid.
Bounded by the paraboloid \(z = 5 + 2x^2 + 2y^2\) and the plane z = 11 in the first octant

asked 2021-09-14

The part of the surface \(\displaystyle{2}{y}+{4}{z}−{x}^{{2}}={5}\) that lies above the triangle with vertices \(\displaystyle{\left({0},{0}\right)},{\left({2},{0}\right)},{\quad\text{and}\quad}{\left({2},{4}\right)}\) Find the area of the surface.

asked 2021-09-07
The plane \(\displaystyle{x}+{y}+{2}{z}={18}\) intersects the paraboloid \(\displaystyle{z}={x}^{{2}}+{y}^{{2}}\) in an ellipse. Find the points on this ellipse that are nearest to and farthest from the origin.
asked 2021-09-01
Write an equation in the indicated coordinate system. Also write a parametric equation and a vector equation for the following equation
\(\displaystyle{x}^{{{2}}}+{y}^{{{2}}}={100}\in{R}^{{{2}}}\), polar coordinates

Plainmath recommends

  • Ask your own question for free.
  • Get a detailed answer even on the hardest topics.
  • Ask an expert for a step-by-step guidance to learn to do it yourself.
Ask Question
...