# Left Endpoint Approximation Calculator for a Table

For the given table of values, the calculator will approximate the integral using left endpoints (the left Riemann sum), with steps shown.

Related calculator: Left Endpoint Approximation Calculator for a Function

## Your Input

**Approximate the integral $$$\int\limits_{-3}^{5} f{\left(x \right)}\, dx$$$ with the left endpoint approximation using the table below:**

$$$x$$$ | $$$-3$$$ | $$$-2$$$ | $$$0$$$ | $$$3$$$ | $$$5$$$ |

$$$f{\left(x \right)}$$$ | $$$-2$$$ | $$$3$$$ | $$$-1$$$ | $$$2$$$ | $$$5$$$ |

## 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, $$$\int\limits_{-3}^{5} f{\left(x \right)}\, dx\approx \left(-2 - \left(-3\right)\right) \left(-2\right) + \left(0 - \left(-2\right)\right) 3 + \left(3 - 0\right) \left(-1\right) + \left(5 - 3\right) 2 = 5.$$$

## Answer

**$$$\int\limits_{-3}^{5} f{\left(x \right)}\, dx\approx 5$$$A**