calculus

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The region bounded by the given curves is rotated about the specified axis. Find the volume V of the resulting solid by any method.
y = −x2 + 7x − 12, y = 0; about the x-axis

  • calculus -

    y = -(x-3)(x-4)

    So, using discs of thickness dx,

    v = ∫[3,4] πr^2 dx
    where r=y
    v = ∫[3,4] π(x^2-7x+12)^2 dx = π/30

    Using shells of thickness dy, and taking advantage of the symmetry,

    v = 2∫[0,1/4] 2πrh dy
    where r=y and h=√(1/4 - y)
    v = 2∫[0,1/4] 2πy√(1/4 - y) dy = π/30

  • calculus -

    A variable force of
    7x−2
    pounds moves an object along a straight line when it is x feet from the origin. Calculate the work done in moving the object from
    x = 1
    ft to
    x = 19
    ft. (Round your answer to two decimal places.)

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