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Math Help - limits of Riemann sum help

  1. #1
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    limits of Riemann sum help

    Hi, i have a question in my hw that i find rather difficult. -

    compute the integrals by finding the limits of the Riemann sums
    integral of x^3 dx
    use summation series

    my textbook is very unclear about how to find the limit of the Riemann sums, so i have no clue what to do there. and what do they mean by summation series?

    thanks
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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by docpar View Post
    Hi, i have a question in my hw that i find rather difficult. -

    compute the integrals by finding the limits of the Riemann sums
    integral of x^3 dx
    use summation series

    my textbook is very unclear about how to find the limit of the Riemann sums, so i have no clue what to do there. and what do they mean by summation series?

    thanks
    We are attempting to find F(x)+C=\int{x^3}dx, ok to do this let t,\xi\in\mathbb{R} and suppose that there exists a constant ( \xi) such that F(\xi)=0. Then

    F(t)=\int_{\xi}^{t}x^3dx.

    So now by definition \int_a^{b}f(x)dx=\lim_{n\to\infty}\sum_{i=1}^{n}f\  left(M_i\right)\Delta{x}

    where \Delta{x}=\frac{b-a}{n} and M_i=a+\left(\Delta{x}\right)i

    So in our case f(x)=x^3, M_i=\xi+\frac{t-\xi}{n}i, \Delta{x}=\frac{t-\xi}{n}

    So \begin{aligned}\int_{\xi}^{t}x^3dx&=\lim_{n\to\inf  ty}\sum_{i=1}^{n}\left(\xi+\frac{t-\xi}{n}i\right)^3\cdot\frac{t-\xi}{n}\end{aligned}

    So just calculate that, note that by a clever observation about polynomials you can show that \xi=0
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