# Homework Help: Volume of bounded region and equilateral triangle

1. Mar 4, 2013

### Painguy

1. The problem statement, all variables and given/known data
y=x^2 y=1

Find the volume of the bounded region using an equilateral triangle cross section

2. Relevant equations

c^2=a^2+b^2

3. The attempt at a solution
I'm will solve it with respect to x 1st.

2∫((1-x^2)h)/2 dx from 0 to 1

base=2(1-x^2)

(2-2x^2)^2=(1-x^2)^2+h^2

4-8 x^2+4 x^4=1-2 x^2+x^4 +h^2
3-6x^2+3x^4=b^2
sqrt(3x^4-6x^2+3)=h

∫(1-x^2)(sqrt(3x^4-6x^2+3))dx
sqrt(3)∫(1-x^2)sqrt((1-x^2)^2) dx
sqrt(3)∫1-2x^2+x^4 dx from 0 to 1

sqrt(3)(x-(2x^3)/3 +(x^5)/5) from 0 to 1
sqrt(3)(1-(2)/3 +(1)/5)
sqrt(3)(8/15)

Is this right?

Here is teh same thing, but with respect to y

-2∫ (sqrt(y)h)/2 dy from 1 to 0

(2sqrt(y))^2=sqrt(y)^2 +h^2
4y=y+h^2
sqrt(3y)=h

-∫sqrt(y)sqrt(3y)dy
-sqrt(3)∫ y dy
-sqrt(3) ((y^2)/2) from 0 to 1
sqrt(3)/2)

Is this right? I know the volumes will be different if i do them with respect to a different axis, but I just wanted to practice both ways.

Last edited: Mar 4, 2013