Riemann Sums

Let $\mathit{f}\left(\mathit{x}\right)$ be defined on a closed interval $\left[\mathit{a},\mathit{b}\right]$ and let P be a regular partition of $\left[\mathit{a},\mathit{b}\right]$. Let $\mathit{\Delta x}$ be the width of each subinterval $\left[{\mathit{x}}_{\mathit{i}-1,}{\mathit{x}}_{\mathit{i}}\right]$and for each $\mathit{i}$, let ${\mathit{x}}_{\mathit{i}}^{\mathit{t}}$ be any point in $\left[{\mathit{x}}_{\mathit{i}-1},{\mathit{x}}_{\mathit{i}}\right]$. A Riemann sum is defined for $\mathit{f}\left(\mathit{x}\right)$ as

$\sum _{\mathit{i}=1}^{\mathit{n}}\mathit{f}\left({\mathit{x}}_{\mathit{i}}^{\mathit{t}}\right)\mathit{\Delta x}$

Area under the curve using Riemann Sums

Let $\mathit{f}\left(\mathit{x}\right)$ be a continuous, nonnegative function on an interval $\left[\mathit{a},\mathit{b}\right]$, and let $\sum _{\mathit{i}=1}^{\mathit{n}}\mathit{f}\left({\mathit{x}}_{\mathit{i}}^{\mathit{t}}\right)\mathit{\Delta x}$ be a Riemann sum for $\mathit{f}\left(\mathit{x}\right)$. Then, the area under the curve $\mathit{y}=\mathit{f}\left(\mathit{x}\right)$ on $\left[\mathit{a},\mathit{b}\right]$ is given by

$\mathit{A}=\underset{\mathit{n}\to \infty }{\lim }\sum _{\mathit{i}=1}^{\mathit{n}}\mathit{f}\left({\mathit{x}}_{\mathit{i}}^{\mathit{t}}\right)\mathit{\Delta x}$

Left Endpoint Approximation

On each subinterval $\left[{\mathit{x}}_{\mathit{i}-1},{\mathit{x}}_{\mathit{i}}\right]$, construct a rectangle with width $\mathit{\Delta x}$ and height equal to $\mathit{f}\left({\mathit{x}}_{\mathit{i}-1}\right)$, which is the function value at the left endpoint of the subinterval. Then the area of this rectangle is $\mathit{f}\left({\mathit{x}}_{\mathit{i}-1}\right)\mathit{\Delta x}$. Adding the areas of all these rectangles, we get an approximate value for $\mathit{A}$. We use the notation L_n to denote that this is a left-endpoint approximation of $\mathit{A}$ using $\mathit{n}$ subintervals.

$\mathit{A}\approx {\mathit{L}}_{\mathit{n}}=\mathit{f}\left({\mathit{x}}_0\right)\mathit{\Delta x}+\mathit{f}\left({\mathit{x}}_1\right)\mathit{\Delta x}+...+\mathit{f}\left({\mathit{x}}_{\mathit{n}-1}\right)\mathit{\Delta x}=\sum _{\mathit{i}=1}^{\mathit{n}}\mathit{f}\left({\mathit{x}}_{\mathit{i}-1}\right)\mathit{\Delta x}$

Right Endpoint Approximation

Construct a rectangle on each subinterval $\left[{\mathit{x}}_{\mathit{i}-1},{\mathit{x}}_{\mathit{i}}\right]$, only this time the height of the rectangle is determined by the function value $\mathit{f}\left({\mathit{x}}_{\mathit{i}}\right)$ at the right endpoint of the subinterval. Then, the area of each rectangle is $\mathit{f}\left({\mathit{x}}_{\mathit{i}}\right)\mathit{\Delta x}$ and the approximation for $\mathit{A}$ is given by:

$\mathit{A}\approx {\mathit{R}}_{\mathit{n}}=\mathit{f}\left({\mathit{x}}_1\right)\mathit{\Delta x}+\mathit{f}\left({\mathit{x}}_2\right)\mathit{\Delta x}+...+\mathit{f}\left({\mathit{x}}_{\mathit{n}}\right)\mathit{\Delta x}=\sum _{\mathit{i}=1}^{\mathit{n}}\mathit{f}\left({\mathit{x}}_{\mathit{i}}\right)\mathit{\Delta x}$

Approximate the area

Riemann sum is a method used to approximate the area under a curve by dividing the region into smaller rectangles and summing their areas. Here's a brief explanation of how it works:

1. Divide the interval of interest into smaller subintervals.
2. Choose a representative point within each subinterval (left, right, or midpoint).
3. Compute the function value at each representative point.
4. Multiply each function value by the width of the corresponding subinterval.
5. Sum up all the products obtained in the previous step.

By following these steps, the Riemann sum provides an approximation of the area under the curve. The accuracy of the approximation increases as the number of subintervals (and consequently, the rectangles) increases, approaching the exact area under the curve as the number of rectangles approaches infinity.