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Find the area of the region bounded by the parabola y = 4x^2, the tangent line to this parabola at (4, 64), and the x-axis.

  • calculous -

    First, find the equation of the tangent line:

    y = 4x^2
    y' = 8x
    slope at (4,64) = 32

    (y-64)/(x-4) = 32
    y = 32x - 64

    32x-64 crosses the x-axis at x=2
    So, we need to break the area up into two parts.

    Area between the curve and y=0 on [0,2]
    Area between curve and tangent line on [2,4]

    Area = Int(4x^2 dx)[0,2] + Int(4x^2 - (32x-64))[2,4]

    = (4/3 x^3)[0,2] + (4/3 x^3 - 16x^2 + 64x)[2,4]

    = [4/3 * 8] + [4/3 * 64 - 16*16 + 64*4] - [4/3 * 8 - 16*4 + 64*2]

    = 64/3

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