Calculus??????????????????????????????

Points A (2,2) and B (-1,-1) are on the parabola y= x^2-2. Placing point P (p, p^2 - 2) on the parabola between points A and B:

Express the area of triangle ABP in terms of p.

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  • 1 decade ago
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    Shift the whole system up by 2 units to make calculations easier.

    The parabola will then be y = x^2, with A = (2, 4), B = (-1, 1) and P = (p, p^2).

    Equation to parabola is : y0 = x^2

    Equation to AB is : y1 = x + 2

    Equation to BP is : y2 = (p - 1)x + p

    Equation to AP is : y3 = (p + 2)x - 2p

    Area of triangle ABP

    = ∫(y1 - y0) dx (from -1 to 2) - ∫(y2 - y0) dx (from -1 to p) - ∫(y3 - y0) dx (from p to 2)

    = ∫(x + 2 - x^2) dx (-1→2) - ∫[(p-1)x + p - x^2] dx (-1→p) - ∫[(p+2)x - 2p - x^2)] dx (p→2)

    = [x^2/2+2x-x^3/3](-1→2) - [(p-1)x^2/2+px-x^3/3](-1→p) - [(p+2)x^2/2-2px-x^3/3](p→2)

    = [(6 - 8/3) - (-7/6)] - [(p^3/6 + p^2/2) - (-p/2 - 1/6)] - [(-2p + 4/3) - (p^3/6 - p^2)]

    = (3/2)(1 + p)(2 - p)

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