Thread: Finding volume of object in R3

1. Finding volume of object in R3

Find te volume of the solid in the first octant of XYZ space bounded below by the coordinate axes and the unit circle, and bounded above by z = 8xy.

So i want the double integral of 8xy dy dx.

The outer integral should be simple, x goes from 0 to 1.

The inner integral though we need to solve for y. y^2 + x^2 = 1.
So y = + or - sqrt(1 - x^2) I only want the positive root.

But since its in the first octant should it be sqrt(1-x^2)/2? because i only want the quarter circle... So my limits for y should be 0 to sqrt(1-x^2)/2.

Do i need to divide by 2?

2. Originally Posted by jstout64
Find te volume of the solid in the first octant of XYZ space bounded below by the coordinate axes and the unit circle, and bounded above by z = 8xy.

So i want the double integral of 8xy dy dx.

The outer integral should be simple, x goes from 0 to 1.

The inner integral though we need to solve for y. y^2 + x^2 = 1.
So y = + or - sqrt(1 - x^2) I only want the positive root.

But since its in the first octant should it be sqrt(1-x^2)/2? because i only want the quarter circle... So my limits for y should be 0 to sqrt(1-x^2)/2.

Do i need to divide by 2?
No, x is bounded by -1 and 1 not from 0-->1. But since we want the first octant, 0--1 is correct but this already sets the bounds for y. Thus, y does not need to be divided by 2.