# 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

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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.

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Chapter 18 / Lesson 12Learn 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.

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