Simpson's Rule Calculator for a Table
Approximate an integral (given by a table of values) using the Simpson's rule step by step
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 calculators: Simpson's Rule Calculator for a Function, Simpson's 3/8 Rule Calculator for a Table
Your Input
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, $$$\int\limits_{0}^{8} f{\left(x \right)}\, dx\approx \frac{2 - 0}{3} \left(-1 + \left(4\right)\cdot \left(5\right) + 0\right) + \frac{6 - 4}{3} \left(0 + \left(4\right)\cdot \left(2\right) + 7\right) = \frac{68}{3}.$$$
Answer
$$$\int\limits_{0}^{8} f{\left(x \right)}\, dx\approx \frac{68}{3}\approx 22.666666666666667$$$A