# Left Endpoint Approximation Calculator for a Table

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