Trapezoidal Rule Calculator for a Table

For the given table of values, the calculator will approximate the integral by means of the trapezoidal rule, with steps shown.

Related calculator: Trapezoidal Rule Calculator for a Function

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.

Your Input

Approximate the integral $$$\int\limits_{1}^{11} f{\left(x \right)}\, dx$$$ with the trapezoidal rule using the table below:

$$$x$$$$$$1$$$$$$3$$$$$$5$$$$$$7$$$$$$9$$$$$$11$$$
$$$f{\left(x \right)}$$$$$$4$$$$$$0$$$$$$-2$$$$$$-3$$$$$$6$$$$$$-5$$$

Solution

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

Therefore, $$$\int\limits_{1}^{11} f{\left(x \right)}\, dx\approx \left(3 - 1\right) \frac{0 + 4}{2} + \left(5 - 3\right) \frac{-2 + 0}{2} + \left(7 - 5\right) \frac{-3 - 2}{2} + \left(9 - 7\right) \frac{6 - 3}{2} + \left(11 - 9\right) \frac{-5 + 6}{2} = 1.$$$

Answer

$$$\int\limits_{1}^{11} f{\left(x \right)}\, dx\approx 1$$$A