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Math Help - integral meaning

  1. #1
    Junior Member
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    integral meaning

    It says "find the area of the region bounded by the graph of f(x)=7x sin(x)
    and the x-axis on teh interval [0,pi]."

    would i do this...?

    integral as 0 goes to pi of 7xsin(x) dx ? or would i have to square it?
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  2. #2
    Math Engineering Student
    Krizalid's Avatar
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    Yes, you have to compute

    \int_0^\pi7x\sin x\,dx=7\int_0^\pi x\sin x\,dx

    The remaining integral requires integration by parts.
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  3. #3
    MHF Contributor
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    Yes. And do not square it.
    Area = w*L
    dA = y*dx = f(x) *dx = 7x*sin(x) *dx
    dx goes from x=0 to x=pi
    So,
    A = (7)INT.(0 --> pi)[x*sin(x)]dx --------------(i)

    (i) cannot be integrated yet because x*dx is not the derivative of sin(x).

    We use integration by parts.

    INT.[u*dv] = u*v -INT.[v*du] --------------***

    Let u = x
    So, du =dx

    And dv = sin(x) *dx
    So, v = -cos(x)

    Then the integrand of (i) becomes
    7(-x*cos(x) -INT[-cos(x)*dx])
    7(-x*cos(x) +INT[cos(x)]dx)

    The boundaries of dx remain the same, from x=0 to x=pi

    So,
    A = 7[-x*cos(x) +sinx]|(0 --> pi)
    A = 7[{-pi*cos(pi) +sin(pi)} -{-0 +0}]
    A = 7[-pi*(-1) +0]
    A = 7pi sq.units ------------------answer.
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