# 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.

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

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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