aprovard

Answered 2021-06-09
Author has **11117** answers

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

asked 2021-09-02

\(4x + 3y + z = 12\)

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

\(x^{2}+y^{2}+z^{2}=8\).

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

\(\displaystyle{z}=−{6}\)

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

asked 2021-09-06

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