# Simpson's Rule Calculator for a Table

For the given table of values, the calculator will find the approximate value of the integral using Simpson's (parabolic) 1/3 rule, with steps shown.

Related calculator: Simpson's Rule Calculator for a Function

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

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Approximate the integral $\int\limits_{0}^{8} f{\left(x \right)}\, dx$ with the Simpson's rule using the table below:

 $x$ $0$ $2$ $4$ $6$ $8$ $f{\left(x \right)}$ $-1$ $5$ $0$ $2$ $7$

## Solution

The Simpson's 1/3 rule approximates the integral using parabolas: $\int\limits_{a}^{b} f{\left(x \right)}\, dx\approx \sum_{i=1}^{\frac{n - 1}{2}} \frac{\Delta x_{i}}{3} \left(f{\left(x_{2i-1} \right)} + 4 f{\left(x_{2i} \right)} + f{\left(x_{2i+1} \right)}\right)$, where $n$ is the number of points and $\Delta x_{i}$ is the length of subinterval no. $2 i - 1$.

$\int\limits_{0}^{8} f{\left(x \right)}\, dx\approx \frac{2 - 0}{3} \left(f{\left(0 \right)} + 4 f{\left(2 \right)} + f{\left(4 \right)}\right) + \frac{6 - 4}{3} \left(f{\left(4 \right)} + 4 f{\left(6 \right)} + f{\left(8 \right)}\right)$

Therefore, $\int\limits_{0}^{8} f{\left(x \right)}\, dx\approx \frac{2 - 0}{3} \left(-1 + \left(4\right)\cdot \left(5\right) + 0\right) + \frac{6 - 4}{3} \left(0 + \left(4\right)\cdot \left(2\right) + 7\right) = \frac{68}{3}.$

$\int\limits_{0}^{8} f{\left(x \right)}\, dx\approx \frac{68}{3}\approx 22.666666666666667$A