Question

asked 2021-06-08

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

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 2021-09-12

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 2021-09-14

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