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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Your Input

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

Answer

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