Find the volume of the solid in the first octant bounded by the coordinate plane

Jaya Legge 2021-11-02 Answered
Find the volume of the solid in the first octant bounded by the coordinate planes, the plane x = 3, and the parabolic cylinder
\(\displaystyle{z}={4}-{y}^{{{2}}}\).

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

Faiza Fuller
Answered 2021-11-03 Author has 9177 answers
\(\displaystyle\int\int_{{{R}}}{f{{\left({x},{y}\right)}}}{d}{A}\)
To find the value of double integral, we must found the area of integration.
It is given:
\(\displaystyle{f{{\left({x},{y}\right)}}}={4}-{y}^{{{2}}}\)
The domain is restricted by the area x=3.
Now, we can solve the integral:
\(\displaystyle{\int_{{{0}}}^{{{2}}}}{\int_{{{0}}}^{{{3}}}}{\left({4}-{y}^{{{2}}}\right)}{\left.{d}{x}\right.}{\left.{d}{y}\right.}={\int_{{{0}}}^{{{2}}}}{{\left[{4}{x}-{y}^{{{2}}}{x}\right]}_{{{0}}}^{{{3}}}}{\left.{d}{y}\right.}\)
\(\displaystyle={\int_{{{0}}}^{{{2}}}}{\left({12}-{3}{y}^{{{2}}}\right)}{\left.{d}{y}\right.}\) (Use \(\displaystyle\int{x}^{{{a}}}{\left.{d}{x}\right.}={\frac{{{x}^{{{a}+{1}}}}}{{{a}+{1}}}}\))
\(\displaystyle={{\left[{12}{y}-{y}^{{{3}}}\right]}_{{{0}}}^{{{2}}}}\)
=24-8
=16
Results: 16
Have a similar question?
Ask An Expert
0
 

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 2021-06-08

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 2021-02-24

Write inequalities to describe the sets The solid cube in the first octant bounded by the coordinate planes and the planes \(x = 2, y = 2, and\ z = 2\)

asked 2021-09-17
Find the volume of the solid in the first octant bounded by the cylinder \(\displaystyle{z}={16}-{x}^{{{2}}}\) and the plane y=5
asked 2021-10-28
Find the volume of the solid that lies inside both of the spheres
\(\displaystyle{x}^{{2}}+{y}^{{2}}+{z}^{{2}}+{4}{x}-{2}{y}+{4}{z}+{5}={0}\)
and
\(\displaystyle{x}^{{2}}+{y}^{{2}}+{z}^{{2}}={4}\)
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 2021-02-25

Determine the volume of the largest box in the first octant with three in the coordinate planes, one vertex at the origin, and its opposite vertex in the plane \(x+3y+5z=15\)

asked 2021-02-24

Find the average value of F(x, y, z) over the given region. \(F(x, y, z) = x2 + 9\) over the cube in the first octant bounded by the coordinate planes and the planes \(x = 2, y = 2, and\ z =2\).

...