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Math Help - area bounded by two curves

  1. #1
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    Exclamation area bounded by two curves

    Ive found the antiderivative to the two functions and im at the last part, where you have to find the total area. This is what it looks like so far...

    [ -(x^3)/3 -(x^2)/2 +2x ] theres a small 1 at the top of the last square bracket and a small -2 at the bottom (interval of [-1,2])

    Which numbers do I sub in to get the final answer??
    I keep thinking its just the first one, but my answer keeps coming out wrong. I keep getting 7/6, but my textbook says the final answer is 9/2.

    Where am I messing up?
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by linearalgebra View Post
    Ive found the antiderivative to the two functions and im at the last part, where you have to find the total area. This is what it looks like so far...

    [ -(x^3)/3 -(x^2)/2 +2x ] theres a small 1 at the top of the last square bracket and a small -2 at the bottom (interval of [-1,2])

    Which numbers do I sub in to get the final answer??
    I keep thinking its just the first one, but my answer keeps coming out wrong. I keep getting 7/6, but my textbook says the final answer is 9/2.

    Where am I messing up?
    What were the two original functions?
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  3. #3
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    Quote Originally Posted by Chris L T521 View Post
    What were the two original functions?
    y= 2-(x^2)
    y=x
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  4. #4
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by linearalgebra View Post
    y= 2-(x^2)
    y=x
    First, find the intersection points

    x=2-x^2\implies x^2+x-2=0\implies\left(x+2\right)\left(x-1\right)=0\implies x=1 or x=-2


    So, the area is \int_{-2}^1\left[\left(2-x^2\right)-x\right]\,dx=\int_{-2}^1 2-x-x^2\,dx=\left.\left[2x-\tfrac{1}{2}x^2-\tfrac{1}{3}x^3\right]\right|_{-2}^1 =\left(2-\tfrac{1}{2}-\tfrac{1}{3}\right)-\left(-4-2+\tfrac{8}{3}\right)=5-\tfrac{1}{2}=\tfrac{9}{2}

    Does this make sense?
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  5. #5
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    Quote Originally Posted by Chris L T521 View Post
    First, find the intersection points

    x=2-x^2\implies x^2+x-2=0\implies\left(x+2\right)\left(x-1\right)=0\implies x=1 or x=-2


    So, the area is \int_{-2}^1\left[\left(2-x^2\right)-x\right]\,dx=\int_{-2}^1 2-x-x^2\,dx=\left.\left[2x-\tfrac{1}{2}x^2-\tfrac{1}{3}x^3\right]\right|_{-2}^1 =\left(2-\tfrac{1}{2}-\tfrac{1}{3}\right)-\left(-4-2+\tfrac{8}{3}\right)=5-\tfrac{1}{2}=\tfrac{9}{2}

    Does this make sense?
    The last bit, right before you give your final answer..you have two parts. the first part is where you sub in x=1 and then you subtract the second part in which you sub in x=-2. am i correct?

    If so, then thank you SO much, you made everything so much clearer
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  6. #6
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by linearalgebra View Post
    The last bit, right before you give your final answer..you have two parts. the first part is where you sub in x=1 and then you subtract the second part in which you sub in x=-2. am i correct?

    If so, then thank you SO much, you made everything so much clearer
    Yes, that's correct. Its a direct application of the Fundamental Theorem of Calculus (Pt. II):

    \int_a^bf\!\left(x\right)\,dx=F\!\left(b\right)-F\!\left(a\right).
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  7. #7
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    awesome. thanks again!
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