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Math Help - Definite integral

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    Definite integral

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

    a)the value of 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 y=4x+1 and the curve y=(2x-3)(x-2) intersecting at point (k,3).Find

    a)the value of k

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

    So 4x +1 = (2x - 3)(x - 2)

    4x + 1 = 2x^2 - 4x - 3x + 6

    4x + 1 = 2x^2 - 7x + 6

    0 = 2x^2 - 11x + 5

    0 = 2x^2 - x - 10x + 5

    0 = x(2x - 1) - 5(2x - 1)

    0 = (2x - 1)(x - 5)

    2x - 1 = 0 or x - 5 = 0

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


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


    So you are wanting to find

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