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