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Math Help - Multi - Step Integral Problem

  1. #1
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    Multi - Step Integral Problem

    Problem
    f(x) = (1-x^4)^\frac{1}{6}\
    R is the area under the curve from 0 to 1.



    Part A: Write down an integral that represents the volume of this solid using the slice method, rotating around the x-axis.


    For this part I got an answer that I am pretty sure is right: pi*\int^1_0 (1-x^4)^\frac{2}{6}\\,dx

    Part B: Write down the integral that represents the volume of this solid using the shell method, rotating around the x-axis.

    For this part I put f(x) in terms of y and I got 2*pi*\int^1_0( y*(1-y^6)^\frac{1}{4}\\,dx

    Part C: The integral you obtain in part (b) should be an integral in y. Now making the substitution y = f(x) show that the integral in part (b) is equal to:
    -\int^1_0(2*pi*x*f(x))\  dfdx/dx

    Using the substitution I got everything but the negative sign on the outside.

    Part C(2): Hence, show that the integral in part (b) is equal to -\int^1_0(pi*x)\(d(f^2)/dx)dx

    I did use using the chain rule, d(f^2) = 2f(x) and then the two integrals are the same.
    Part D: Using part (c), show that the integral in part (b) is equal to the integral in part (a).

    No idea what to do

    Thanks so much for the help
    Last edited by auntjamima; October 19th 2010 at 10:21 PM.
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