# Simpson's 3/8 Rule Calculator for a Table

For the given table of values, the calculator will find the approximate value of the integral using Simpson's 3/8 rule, with steps shown.

Related calculators: Simpson's Rule Calculator for a Table, Simpson's 3/8 Rule Calculator for a Function

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

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Approximate the integral $\int\limits_{0}^{12} f{\left(x \right)}\, dx$ with the Simpson's 3/8 rule using the table below:

 $x$ $0$ $2$ $4$ $6$ $8$ $10$ $12$ $f{\left(x \right)}$ $5$ $-2$ $1$ $6$ $7$ $3$ $4$

## Solution

The Simpson's 3/8 rule approximates the integral using cubic polynomials: $\int\limits_{a}^{b} f{\left(x \right)}\, dx\approx \sum_{i=1}^{\frac{n - 1}{3}} \frac{3 \Delta x_{i}}{8} \left(f{\left(x_{3i-2} \right)} + 3 f{\left(x_{3i-1} \right)} + 3 f{\left(x_{3i} \right)} + f{\left(x_{3i+1} \right)}\right)$, where $n$ is the number of points and $\Delta x_{i}$ is the length of subinterval no. $3 i - 2$.

$\int\limits_{0}^{12} f{\left(x \right)}\, dx\approx \frac{3 \left(2 - 0\right)}{8} \left(f{\left(0 \right)} + 3 f{\left(2 \right)} + 3 f{\left(4 \right)} + f{\left(6 \right)}\right) + \frac{3 \left(8 - 6\right)}{8} \left(f{\left(6 \right)} + 3 f{\left(8 \right)} + 3 f{\left(10 \right)} + f{\left(12 \right)}\right)$

Therefore, $\int\limits_{0}^{12} f{\left(x \right)}\, dx\approx \frac{3 \left(2 - 0\right)}{8} \left(5 + \left(3\right)\cdot \left(-2\right) + \left(3\right)\cdot \left(1\right) + 6\right) + \frac{3 \left(8 - 6\right)}{8} \left(6 + \left(3\right)\cdot \left(7\right) + \left(3\right)\cdot \left(3\right) + 4\right) = 36.$

$\int\limits_{0}^{12} f{\left(x \right)}\, dx\approx 36$A