Simpson's Rule Calculator for a Table

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

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Your Input

Approximate the integral $$$\int\limits_{0}^{6} f{\left(x \right)}\, dx$$$ with the Simpson's rule using the table below:

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

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. $$$2 i - 1$$$.

$$$\int\limits_{0}^{6} f{\left(x \right)}\, dx\approx \frac{2 - 0}{3} \left(f{\left(0 \right)} + 4 f{\left(2 \right)} + f{\left(4 \right)}\right) + \frac{5 - 4}{3} \left(f{\left(4 \right)} + 4 f{\left(5 \right)} + f{\left(6 \right)}\right)$$$

Therefore, $$$\int\limits_{0}^{6} f{\left(x \right)}\, dx\approx \frac{2 - 0}{3} \left(-1 + \left(4\right)\cdot \left(5\right) + 0\right) + \frac{5 - 4}{3} \left(0 + \left(4\right)\cdot \left(2\right) + 7\right) = \frac{53}{3}.$$$

Answer

$$$\int\limits_{0}^{6} f{\left(x \right)}\, dx\approx \frac{53}{3}\approx 17.6666666666667$$$A