# Riemann Sum Calculator for a Table

For the given table of values, the calculator will approximate the definite integral using the Riemann sum and the sample points of your choice: left endpoints, right endpoints, midpoints, and trapezoids.

Related calculator: Riemann Sum Calculator for a Function

 $x$ $f{\left(x \right)}$

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

Approximate the integral $\int\limits_{0}^{8} f{\left(x \right)}\, dx$ with the left Riemann sum using the table below:

 $x$ $0$ $2$ $4$ $6$ $8$ $f{\left(x \right)}$ $1$ $-2$ $5$ $0$ $7$

## Solution

The left Riemann sum approximates the integral using left endpoints: $\int\limits_{a}^{b} f{\left(x \right)}\, dx\approx \sum_{i=1}^{n - 1} \left(x_{i+1} - x_{i}\right) f{\left(x_{i} \right)}$, where $n$ is the number of points.

Therefore, $\int\limits_{0}^{8} f{\left(x \right)}\, dx\approx \left(2 - 0\right) 1 + \left(4 - 2\right) \left(-2\right) + \left(6 - 4\right) 5 + \left(8 - 6\right) 0 = 8$.

$\int\limits_{0}^{8} f{\left(x \right)}\, dx\approx 8$A