calculus

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given the graph of f(x) = x sinx, 0<=x<=2pi
assuming that a quantity y changes at a rate of y' = xsinx, find by how much it will increase or decrease over 3pi/2 <= x <= 2pi

  • calculus -

    This is just a problem in integrals. We have y', and we want to add up all the small changes given by that function.

    Int[x sinx dx] can be solved using integration by parts.

    Recall the product rule of derivatives:

    (uv)' = u'v + uv'
    so
    uv' = (uv)' - u'v

    Going the other direction,

    Int[u dv] = uv - Int[v du]

    So, we want to split up the integrand into two factors, where one part (u) gets simpler after differentiation, and the other part (dv) can be easily integrated.

    Here, we have x sinx
    If u = x, then du = dx
    If dv = sinx, then v = -cosx

    Int[x sinx dx] = uv - Int[v du]
    = -x cosx - Int[-cos x * dx]

    Now we have a simple integrand, -cosx

    Int cos x dx = sin x
    So, the final integration is

    Int[x sinx dx] = uv - Int[v du]
    = -x cosx - Int[-cos x * dx]
    = -x cosx + sin x

    Evaluating at 3pi/2 and 2pi, we have

    (-2pi * 1 + 0) - (-3pi/2 * 0 + -1) = -2pi + 3pi/2 + 1 = 1 + 7pi/2

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