Calculus

The curve y = 11x - 24 - x^2 cuts the x-axis at points A and B, and PN is the greatest positive value of the y coordinate. Show that 2 PN • AB equals three times the area bounded by that portion of the curve which lies in the first quadrant.

Kinda confused on how to do this

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  1. well, let's find the vertex and intercepts.
    x^2 - 11 x = -y - 24

    x^2 - 11 x + 121/4 = -y - 24 + 30.25

    (x-11/2)^2 = -(y-6.25)
    so PN = 6.25
    now what are A and B
    when y = 0
    (x-11/2)^2 = 6.25
    x-5.5 = +/- 2.5
    x = 3 or 8
    A is at 3 and B is at 8
    so 2 PN * AB = 62.5
    divide by three = 20.83333.....
    now the integral
    from 3 to 8
    of (-x^2 +11 x - 24)dx
    -x^3/3 + 11x^2/2 - 24 x
    at 8
    -10 2/3
    at 3
    -31.5
    20.8333... Whew !

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