# Midpoint Rule Calculator for a Table

## Approximate an integral (given by a table of values) using the midpoint rule step by step

For the given table of values, the calculator will approximate the integral using the midpoint rule, with steps shown.

Related calculator: Midpoint Rule Calculator for a Function

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

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Approximate the integral $\int\limits_{-4}^{4} f{\left(x \right)}\, dx$ with the midpoint rule using the table below:

 $x$ $-4$ $-2$ $0$ $2$ $4$ $f{\left(x \right)}$ $1$ $2$ $7$ $5$ $3$

### Solution

The midpoint rule approximates the integral using midpoints: $\int\limits_{a}^{b} f{\left(x \right)}\, dx\approx \sum_{i=1}^{\frac{n - 1}{2}} \left(x_{2i+1} - x_{2i-1}\right) f{\left(\frac{x_{2i-1} + x_{2i+1}}{2} \right)}$, where $n$ is the number of points.

$\int\limits_{-4}^{4} f{\left(x \right)}\, dx\approx \left(0 - \left(-4\right)\right) f{\left(\frac{0 - 4}{2} \right)} + \left(4 - 0\right) f{\left(\frac{4 + 0}{2} \right)}$

$\int\limits_{-4}^{4} f{\left(x \right)}\, dx\approx \left(0 - \left(-4\right)\right) f{\left(-2 \right)} + \left(4 - 0\right) f{\left(2 \right)}$

Therefore, $\int\limits_{-4}^{4} f{\left(x \right)}\, dx\approx \left(0 - \left(-4\right)\right) 2 + \left(4 - 0\right) 5 = 28$.

$\int\limits_{-4}^{4} f{\left(x \right)}\, dx\approx 28$A