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Volume of bounded region and equilateral triangle

  1. Mar 4, 2013 #1
    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
  2. jcsd
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