# Midpoint Rule Calculator for a Table

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

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

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

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