# Thread: limits of Riemann sum help

1. ## 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

2. Originally Posted by docpar
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 $\displaystyle F(x)+C=\int{x^3}dx$, ok to do this let $\displaystyle t,\xi\in\mathbb{R}$ and suppose that there exists a constant ($\displaystyle \xi$) such that $\displaystyle F(\xi)=0$. Then

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

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

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

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

So \displaystyle \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 $\displaystyle \xi=0$