- #1
Painguy
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Homework Statement
y=x^2 y=1
Find the volume of the bounded region using an equilateral triangle cross section
Homework Equations
c^2=a^2+b^2
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.
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