Thread: Riemann Sums

I have been trying to figure out how to convert the attached graph into a Riemann sum and then converting that into an integral. I know that the equations are:

$\displaystyle y = \sqrt(x)$ and y = x

I keep getting caught up with the term Dy and am not sure what to do. All help is much appreciated.

Originally Posted by Spudwad

I have been trying to figure out how to convert the attached graph into a Riemann sum and then converting that into an integral. I know that the equations are:

$\displaystyle y = \sqrt(x)$ and y = x

I keep getting caught up with the term Dy and am not sure what to do. All help is much appreciated.

It is not always very easy to use the definition of the Reimann sums for the computation of definite integrals. We rather prefer using the anti-derivatives of fucntion for this job, i.e., we try to find a function $\displaystyle F$ such that $\displaystyle F^{\prime}=f$ on $\displaystyle [a,b]$, and then compute $\displaystyle F(b)-F(a)$, which gives $\displaystyle \textstyle\int_{a}^{b}f(x)\mathrm{d}x$. The computation of the Reimann sum requires in general a high knowledge of sum-difference equations theory.

For instance, you may take a look in the following post, where I have explained the way for the computation of $\displaystyle \textstyle\int_{a}^{b}x^{k}\mathrm{d}x$ (for $\displaystyle k\in\mathbb{N}$) by means of the Reimann sum (post by me). You may study the following post, for computing the Reimann sum for $\displaystyle \textstyle\int_{a}^{b}\log(x)\mathrm{d}x$ (post by Laurent).

It seems hard to do that for $\displaystyle \textstyle\int_{a}^{b}x^{\alpha}\mathrm{d}x$ when $\displaystyle \alpha\in\mathbb{R}^{+}\backslash\mathbb{N}$ (as in your case).

bkarpuz