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Math Help - calculate area of shaded region

  1. #1
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    calculate area of shaded region

    firstly, if I want to find the points on intersection on this graph between the curve and line, I'd set thier y values equal to each other.

     y = x^3

     y = -x+2

     x^3 = -x +2

     x^3 +x -2 = 0

    how do I solve that?

    or do I not need these points when working out the area ?
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  2. #2
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    x^3 = 2 - x @ x = 1

    So from 0 to 1, we have the area under x^3. And from 1 to 2, we have area under 2 - x.

    \int_{0}^{1} x^3dx + \int_{1}^{2} (2 - x)dx
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  3. #3
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    Quote Originally Posted by derfleurer View Post
    x^3 = 2 - x @ x = 1

    \int_{0}^{1} x^3dx + \int_{1}^{2} (2 - x)dx
    How did you work out x=1?
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  4. #4
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    Just a straight forward solution. Couldn't really tell you a method for finding it (maybe someone else can).

    (1)^3 = 2 - (1)
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  5. #5
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    Rational root theorem - Wikipedia, the free encyclopedia

    In this case, we can easily see that x=1 is the root we're looking for.
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  6. #6
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    Can you factor it like this;

     x^3 +x -2 = 0

     x(x^2+1)-2= 0

     x-2 = 0

     (x+1)(x-1) = 0

     x = 2 , x =-1 , x = 1

    ? Is that a valid proccess?
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  7. #7
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    The typical process for finding all roots is to first guess one root using the theorem I linked to above, and then use polynomial long division to factor the polynomial.

    For example, if one root is x=1, then you divide the original polynomial with x-1.

    And no, your process isn't correct. x=2 and x=-1 aren't roots to the posted polynomial.
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