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Thread: Definite integral

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    Definite integral

    The diagram above shows the straight line$\displaystyle y=4x+1$ and the curve $\displaystyle y=(2x-3)(x-2)$ intersecting at point $\displaystyle (k,3)$.Find

    a)the value of $\displaystyle k$

    b)the area of the shaded region.
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  2. #2
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    Quote Originally Posted by mastermin346 View Post
    The diagram above shows the straight line$\displaystyle y=4x+1$ and the curve $\displaystyle y=(2x-3)(x-2)$ intersecting at point $\displaystyle (k,3)$.Find

    a)the value of $\displaystyle k$

    b)the area of the shaded region.
    The graphs intersect where they are equal.

    So $\displaystyle 4x +1 = (2x - 3)(x - 2)$

    $\displaystyle 4x + 1 = 2x^2 - 4x - 3x + 6$

    $\displaystyle 4x + 1 = 2x^2 - 7x + 6$

    $\displaystyle 0 = 2x^2 - 11x + 5$

    $\displaystyle 0 = 2x^2 - x - 10x + 5$

    $\displaystyle 0 = x(2x - 1) - 5(2x - 1)$

    $\displaystyle 0 = (2x - 1)(x - 5)$

    $\displaystyle 2x - 1 = 0$ or $\displaystyle x - 5 = 0$

    $\displaystyle x = \frac{1}{2}$ or $\displaystyle x = 5$.


    Clearly the point you are referring to is $\displaystyle (x, y) = \left(\frac{1}{2}, 3\right)$.


    So you are wanting to find

    $\displaystyle \int_0^{\frac{1}{2}}{2x^2 - 7x + 6\,dx} - \int_0^{\frac{1}{2}}{4x + 1\,dx}$.
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