Find the volume of the solid E in the first octant bounded by the paraboloid z =...
Question:
Find the volume of the solid E in the first octant bounded by the paraboloid {eq}z = \frac{1}{2}(x^2+y^2) {/eq}and the cone {eq}z = \sqrt{2x^2+2y^2} {/eq} and coordinate planes.
Finding the Volume of the Solid :
We need to find the volume of the solid {eq}E {/eq} in the first octant bounded by the paraboloid and the cone. The equation of cone is given by,
{eq}x^2+y^2=r^2 {/eq}
The Volume of the solid is,
{eq}Volume = \int_a^b \int_c^d (z_1 - z_2) dA {/eq}
Answer and Explanation: 1
Become a Study.com member to unlock this answer! Create your account
View this answer{eq}z_{1} = \frac{1}{z} \left ( x^{2} + y^{2} \right ) \\ z_{1} = \sqrt{2x^{2} + 2y^{2}} {/eq}
The intersection of paraboloid, cone is a circle.
{...
See full answer below.
Ask a question
Our experts can answer your tough homework and study questions.
Ask a question Ask a questionSearch Answers
Learn more about this topic:
Get access to this video and our entire Q&A library
How to Calculate the Volume of a Cube: Formula & Practice
from
Chapter 18 / Lesson 12
651K
Learn how to find the volume of a cube. View the formula for calculating the volume of a cube. See examples of how to calculate the volume of a cube using its side length, surface area, or diagonals.
Related to this Question
- Find the volume of the solid in the first octant bounded above the cone z = 1 - sqrt(x^2 + y^2), below by the x, y-plane, and on the sides by the coordinate planes.
- A) Find the volume of solid E in the first octant bounded by the paraboloid z = (x^2 + y^2)/2 and the cone z = sqrt(2x^2 + 2y^2) and the coordinate planes. B) Find the volume of the solid E bounded by
- Find the volume of the solid in the first octant bounded by the sphere [{MathJax fullWidth='false' \rho = 10 }], the coordinate planes, and the cones \emptyset = \frac{\pi}{6} and \emptyset = \
- Find the volume of the solid in the first octant bounded by the sphere rho =20, the coordinate planes, and the cones phi= { pi}{6} and phi= { pi}{3}.
- Find the volume of the solid in the first octant bounded by the coordinate planes, the cylinder x^2 + y^2 = 2^2 and the plane x+z=4
- Find the volume of the solid in the first octant bounded by the coordinate planes, the cylinder x^2 + y^2 = 3 and the plane z + y = 3.
- Find the volume of the solid in the first octant bounded by the coordinate planes, the cylinder x^2 + y^2 = 4, and the plane z + y = 3.
- Find the volume of the solid in the first octant bounded by the coordinate planes, the cylinder x^2+y^2=4, and the plane y+z=3 using rectangular coordinates.
- Determine the volume of the solid in the first octant bounded above by the cone z = 1 - \sqrt{x^2 + y^2} , below by the xy-plane, and on the sides by the coordinate planes.
- Find the volume of the solid in the first octant bounded by the cylinder z = 25 - x^2 and the plane y = 7.
- Find the volume of the solid in the first octant bounded by the cylinder z = 16 - x2 and the plane y =5.
- Use polar coordinates to find the volume of the given solid bounded by the paraboloid z = 6 + 2x^2 + 2y^2 and the plane z = 12 in the first octant
- Find the volume of the solid in the first octant bounded by the planes 2x+z=2 , y+z=2 and the coordinate planes.
- Find the volume of the solid where the solid is bounded below the sphere x^2 + y^2 + z^2 = 4 and above the cone z = \sqrt {x^2 + y^2} in the first octant.
- Find the volume of the solid in the first octant bounded by the plane 4x+2y+2z = 4 and the coordinate planes.
- Find the volume of the solid bounded in the first octant bounded also by the vertical planes x=1 and y=3x, and the paraboloid z=16-x^2-y^2.
- Find the volume of the solid in the first octant bounded by the parabolic cylinder z = 25 - x^2 and the plane y = 4.
- Find the volume of the solid in the first octant bounded by the parabolic cylinder z = 16 - x^2 and the plane y = 4.
- Find the volume of the solid in the first octant bounded by the parabolic cylinder z = 16 - x^2 and the plane y = 5.
- Find the volume of the solid in the first octant bounded by the parabolic cylinder z = 16 - x^2 and the plane y = 1.
- Find the volume of the solid in the first octant bounded by the parabolic cylinder z = 16 - x2 and the plane y = 2.
- Find the volume of the solid in the first octant bounded by the parabolic cylinder z = 25 - x^2 and the plane y = 2.
- Find the volume of the solid in the first octant bounded by the parabolic cylinder z = 16 - x^2 and the plane y = 2.
- Find the volume of the solid in the first octant bounded by the parabolic cylinder z = 25 - x^2 and the plane y = 1.
- Find the volume of the solid in the first octant bounded by the sphere \rho = 22, the coordinate planes, and the cones \phi = \pi / 6 and \phi = \pi / 3 Upper V equals
- Find the volume of the solid in the first octant bounded by the z = 2 - x^2 and the planes z = 0, y = x, y = 0.
- Find the volume of the solid in the first octant bounded by the sphere \rho =16 the coordinate planes, and the cones \phi = x/6 and \phi = x/3.
- Find the volume of the solid in the first octant bounded by the coordinate planes, the plane x = 3, and the parabolic cylinder z = 4 - y^2.
- Find the volume of the solid in the first octant bounded by the coordinate planes, the cylinder x^2 + y^2 = 4, and the plane y + z = 3 using: A) rectangular coordinates. B) polar coordinates.
- Find the volume of the solid in the first octant bounded by the coordinate planes and the surface z=1-y-x^2. Use double integrals.
- Determine the volume of the solid in the first octant bounded by the coordinate planes, the cylinder x^2 + y^2 = 4, and the plane y + z = 3 using rectangular coordinates.
- Use polar coordinates to find the volume of the solid bounded by the paraboloid z = 7 + 2x^2 + 2y^2 and the plane z=13 in the first octant.
- Use polar coordinates to find the volume of the solid bounded by the paraboloid 2 + 2x^2 +2y^2 and the plane z = 8 in the first octant.
- Use polar coordinates to find the volume of the solid bounded by the paraboloid z = 1 = 2x^2 + 2y^2 and the plane z =7 in the first octant.
- Use polar coordinates to find the volume of the solid bounded by the paraboloid z= 10+2x^2+2y^2 and the plane z=16 in the first octant.
- Find the volume of the solid E in the first octant bounded by the paraboloid z = (1/2)(x^2+y^2) and the cone z = sqrt(2x^2+2y^2) and coordinate planes.
- Determine the volume of the solid in the first octant that is bounded by the coordinate planes and the plane x + y/2 + z/4 = 1
- Find the volume of the solid (Use rectangular coordinates). In the first octant bounded by x^2 + z = 64, 3x + 4y = 24, and the 3 - coordinate planes.
- Compute the volume of the solid in the first octant bounded by the coordinate planes, the cylinder x^2 + y^2 = 4, and the plane y + z = 3 using rectangular coordinates.
- Find the volume V of the solid in the first octant that is bounded by the paraboloid z = 4 - x^2 - y^2 and the xy-plane.
- Find the volume of the solid bounded in the first octant by the cylinder z = 9-y^2 and the plane x = 1.
- Find the volume of: A) the solid above the paraboloid z = x^2 + y^2 and below the cone z = sqrt(x^2 + y^2). B) the solid bounded by x^2 + y^2 = 4, y + z = 3 and the xy-plane.
- Find the volume of the solid in the first octant, enclosed by the cylinder z = 9 - y^2 and plane x = 5.
- Find the volume of the solid bounded by the cylinder y^2 + z^2 = 4 and the planes x = 2y, x = 0, z = 0 in the first octant.
- Find the volume of the solid bounded by the cylinder y^2 + z^2 = 9 and the planes x = 2y, x = 0, z = 0 in the first octant.
- Find the volume of the solid bounded by the cylinder y^2 + z^2 = 4 and the planes x = 2 y, x = 0, z = 0 in the first octant.
- 1. Find the volume of the solid under the surface z = 4x + 3y^2 and above the region bounded by x = y^2 and x = y^3. 2. Find the volume of the solid enclosed by the paraboloid z = 3x^2 + y^2 and the planes x = 0, y = 1, y = x, z = 0.
- Use polar coordinates to find the volume of the given solid: Bounded by the paraboloid z = 1 + 2x^2 + 2y^2 and the plane z = 7 in the first octant.
- Determine the volume of the solid in the first octant bounded by the parabolic cylinder z = 16 - x^2 and the plane y = 2.
- Use polar coordinates to find the volume of the given solid. Bounded by the paraboloid z = 2 + 2x^2 + 2y^2 and the plane z = 8 in the first octant.
- Calculate the volume of the solid in the first octant bounded by the coordinate planes, the cylinder x^2 + y^2 = 4, and the plane z + y = 3.
- Find the volume in the first octant bounded by the curve x = 6 - y^2 - z and the coordinate planes.
- Use the polar coordinates to find the volume of the solid bounded by the paraboloid z = 2 + 3x^2 + 3y^2 and the plane z = 14 in the first octant.
- Find the volume of the solid bounded by the paraboloid z = x^2 + y^2 and the plane z = 9 in rectangular coordinates.
- Find the volume of the solid region below the paraboloid z = x ^2 + y ^2 +1, above the coordinate plane z = 0, and within the cylinder x^2 + y^2 = 9.
- Find the volume of the given solid. Bounded by the cylinder y^2 + z^2 = 4 and the planes x = 2y, x = 0, z = 0 in the first octant.
- Find the volume of the given solid. Bounded by the cylinder y^2 + z^2 = 64 and the planes x = 2y, x = 0, z = 0 in the first octant.
- Find the volume of the given solid. Bounded by the cylinder y^2 + z^2 = 64 and the planes x = 2y, x = 0, z = 0 in the first octant.
- Use polar coordinates to find the volume of the given solid which is bounded by the paraboloid, z = 9 + 2x^2 + 2y^2 and the plane, z = 15 in the first octant.
- a. Find the volume of the solid in the first octant bounded by the coordinate planes and the graphs of the equations z = x 2 + y 2 + 1 and 2 x + y = 2 b. If F = 3 u i + u 2 j + ( u + 2 ) k and
- Find the volume of the solid in the first octant bounded by the cylinder z - 4 - y^2 and the plane x - 6. __1. 32 2. 30 3. 40 4. 21 5. 24__
- Find the volume of the solid in the first octant bounded by the cylinder z = 16-x^2 and the plane y=4
- Find the volume of the solid in the first octant bounded by the cylinder z=4-x^2 and the plane y=5
- Find the volume of the solid in the first octant bounded by the cylinder, z = 16 - x^2 and the plane, y = 5.
- Find the volume of the solid in the first octant bounded by the cylinder z= 9-y^2 and the plane x=1 .
- Find the volume of the solid enclosed by the paraboloid z = 3 + x^2 + (y 2)^2 and the planes z = 1, \ x = 2,\ x = 2,\ y = 0, \text{ and } y = 3 .
- Find the volume of the region bounded by the paraboloid z = {7} / {4} + {1} / {4} (x^2 + y^2) and the plane z = 4 that lies outside the cone z^2 - 4 x^2 - 4 y^2 = 0.
- Find the volume of the solid in the first octant bounded by the parabolic cylinder z = 4 - y^2 and the planes x = 0, y = 0, z = 0, and x = 3.
- Find the volume of the solid in the first octant, which is above the cone z = 2*sqrt(x^2 + y^2) and below the sphere x^2 + y^2 + z^2 = 5.
- Compute the volume of the solid bounded by the cylinder y^2 + z^2 = 1 and the planes x = 2y, x = 0, z = 0 in the first octant.
- Find the volume of the solid that is bounded by the single cone z = sqrt(x^2 + y^2) and the paraboloid z = 3 - (x^2)/4 - (y^2)/4.
- Find the volume of the solid in the first octant bounded by the coordinate plane and the graphs of the equations z = x^2 + y^2 +1 and 2x + y = 2.
- Find the volume of the solid in the first octant bounded by the coordinate plane and the graphs of the equations z = x^2 + y^2 +1 \enspace and \enspace 2x + y = 2
- Find the volume of the solid in the first octant bounded by the coordinate plane and the graphs of the equations z = x^2 + y^2 +1 \enspace and \enspace 2x + y = 4
- Find the volume of the solid bounded by the paraboloid z = x^2 + y^2 + 4 and the planes x = 0, y = 0 , z = 0 ,x + y = 1.
- Find the volume of the solid in the first octant that is bounded by the plane y + z = 4, and the y = x^2, and the xy and yz planes.
- Find the volume of the given solid region in the first octant bounded by the plane 2x + 8y +4z = 8 and the coordinate planes, using triple integrals.
- Find the volume of the given solid region in the first octant bounded by the plane 4 x + 5 y + 20 z = 20 and the coordinate planes, using triple integrals
- Find the volume of the given solid region in the first octant bounded by the plane 10x+10y+25z=50 and the coordinate? planes, using triple integrals. solid is x=(5,0,0), y=(0,5,0), z=(0,0,2)
- Find the volume of the solid bounded by the cylinder x^2 + z^2 = 9 and the planes y = 2 x, y = 0, z = 0 in the first octant.
- Find the volume of the solid bounded by the cylinder x^2 + y^2 = 4, and the planes y=z,\ x=0, and z=0 in the first octant.
- Find the volume of the solid bounded by the cylinder x^2 + y^2 = 9 and the planes y = 2z, x = 0, z = 0 in the first octant.
- Find the volume of the solid bounded by the cylinder y ^2 + z ^2 = 1 and the planes x = 2y, x = 0, z = 0 in the first octant.
- Find the volume of the solid in the first octant bounded by the planes x = 0,\ y = 0,\ z = 0,\ x + 2y + z = 0.
- Find the volume of the solid bounded by the paraboloid z = x^2 + y^2 and the plane z = 9.
- Find the volume of the solid bounded by the paraboloid given by z = x^2 + y^2 and the plane z = 9.
- Find the volume of the solid bounded by the paraboloid z = 1 - x^2 - y^2 and the plane z=0.
- Find the volume of the solid bounded by the paraboloid z = x^2 + y^2 and the plane z = 1.
- Find the volume of the solid bounded by the paraboloid z = 6x^2 + 6y^2 and the plane z = 24.
- Find the volume of the solid bounded by the paraboloid z = 6x^2+6y^2 and the plane z=54
- Find the volume of the solid bounded by the planes x = 4, y = 2 , the coordinate planes, and the elliptic paraboloid z = 2 + (x - 1)^2 + 8y^2 .
- Compute the volume of the following solid. The solid in the first octant bounded by the coordinate planes, the cylinder x^2 + y^2 = 4 and the plane z + y = 3.
- Find the volume of the solid bounded by the plane z = 0 and the paraboloid z = 1 - x^2 - y^2.
- Find the volume of the solid bounded by the plane z=10 and the paraboloid z=1+z^2+y^2
- Find the volume of the solid bounded by the plane x = 3 and the paraboloid x = (1/3)y^2 + (1/3)z^2.
- Find the volume of the given solid region in the first octant bounded by the plane 4x+2y+2z=4 and the coordinate planes, using triple intergrals.
- Compute the volume of the solid in the first octant bounded by the parabolic cylinder z = 25 - x^2 and the plane y = 1.
- Compute the volume of the solid in the first octant bounded by the parabolic cylinder z = 16 - x^2 and the plane y = 2.
- Find the volume of the solid in the first octant bounded by the graphs of z = sqrt(x^2 + y^2), and the planes z = 1, x = 0, and y = 0.
- Using a triple integral find the volume of the solid in the first octant bounded by the plane z=4 and the paraboloid z=x^2+y^2.
Explore our homework questions and answers library
Browse
by subject