Trapezoidal Rule Calculator for a Table
Approximate an integral (given by a table of values) using the trapezoidal rule step by step
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
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