# Left Endpoint Approximation Calculator for a Table

## Approximate an integral (given by a table of values) using the left endpoints step by step

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

Related calculator: Left Endpoint Approximation Calculator for a Function

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

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

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