calculus

Find the volume of the solid obtained by rotating the region under the graph f(x) = x2 - 3x about the x-axis over the interval [0, 3].

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  1. What you have is the arch of the parabola, rotated around the x-axis. They conveniently chose [0,3] because f(x) = x(x-3), which has zeroes at 0 and 3.

    Since we have a solid chunk, discs are the way to go.

    Each disc has area pi r^2 = pi * y^2 = pi * (x^4 - 6x^2 + 9)

    Just add up all the discs from 0 to 3. In other words, integrate between the limits of the interval.

    V = Int[0:3] pi * (x^4 - 6x^2 + 9) dx
    = pi(1/5 x^5 - 2 x^3 + 9x)[0:3]

    at x=0, value is 0, so the answer is just the value at x=3:

    pi(243/5 - 54 + 27) = 108/5 pi

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