1) Find the area of the part of the plane4x + 3y + z = 12that 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

Emeli Hagan 2021-06-08 Answered

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

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

aprovard
Answered 2021-06-09 Author has 11117 answers
I have resolved your question. The answer is below:

image

Have a similar question?
Ask An Expert
46
 
content_user
Answered 2021-09-30 Author has 2252 answers

1) \(4x+3y+3=12\)

\(\vec{R}(x,y)=<x,y,12-4x-3y>\)

\(\Rightarrow|\vec{R_x}\times\vec{R_y}|=\sqrt{(f_x)^2+(f_y)^2+1}=\sqrt{(-4)^2+(-3)^2+1}\)

\(=\sqrt{16+9+1}=\sqrt{26}\)

\(\int_S\int ds=\int\int_D\sqrt{26}dA\)

\((x,y,3)\) axis \(=(3,0,0)(0,4,0)(0,0,12)\)

So, the domain D in triangle is bounded by \((0,0)(3,0)(0,4)\)

\(\int\int_D\sqrt{26}dA=\sqrt{26}\times\frac{1}{2}\times3\times4^2=6\sqrt{26}\)

\(z=5+2x^2+2y^2\)

\(\Rightarrow11=5+2x^2+2y^2\)

\(\Rightarrow6=2x^2+2y^2\Rightarrow x^2+y^2=3\ |x^2+y^2=R^2\)

Volume \(=\int_0^{\frac{\pi}{2}}\int_0^{\sqrt{3}}(5+2x^2+2y^2)dA\)

\(=\int_0^{\frac{\pi}{2}}\int_0^\sqrt{3}(5+2R)RdRdQ=\int_0^{\frac{\pi}{2}}\int_0^{\sqrt{3}}3R+2R^2drdQ\)

\(=\int_0^{\frac{\pi}{2}}[\frac{5R^2}{8}+\frac{2R^3}{3}]_0^{\sqrt{3}}dQ=\int_0^{\frac{\pi}{2}}\frac{15}{2}+\frac{6}{\sqrt{3}}dQ\)

\(\Rightarrow\frac{15\pi}{4}+\sqrt{3}\pi=\pi[\frac{15}{4}+\sqrt{3}]\) Answer

10

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 2021-09-02

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 2021-06-09
Change from rectangular to cylindrical coordinates. (Let \(r\geq0\) and \(0\leq\theta\leq2\pi\).)
a) \((-2, 2, 2)\)
b) \((-9,9\sqrt{3,6})\)
c) Use cylindrical coordinates.
Evaluate
\(\int\int\int_{E}xdV\)
where E is enclosed by the planes \(z=0\) and
\(z=x+y+10\)
and by the cylinders
\(x^{2}+y^{2}=16\) and \(x^{2}+y^{2}=36\)
d) Use cylindrical coordinates.
Find the volume of the solid that is enclosed by the cone
\(z=\sqrt{x^{2}+y^{2}}\)
and the sphere
\(x^{2}+y^{2}+z^{2}=8\).
asked 2021-09-12

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

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 2021-08-22
Consider the curve in the plane given in polar coordinates by \(\displaystyle{r}={4}{\sin{\theta}}\). Find the cartesian equation for the curve and identify the curve
asked 2021-01-17

Find the solution of limit \(\displaystyle\lim_{{{\left({x},{y}\right)}\rightarrow{0},{0}}}\frac{{\sqrt{{{x}^{2}+{y}^{2}}}}}{{{x}^{2}+{y}^{2}}}\) by using the polar coordinates system.

asked 2021-09-06

Find the area of the part of the plane \(5x + 3y + z = 15\) that lies in the first octant.

...