# 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

## Your Input

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