# 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.

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