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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

A
xx
f(x)f{\left(x \right)}

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Your Input

Approximate the integral ∫08f(x) dx\int\limits_{0}^{8} f{\left(x \right)}\, dx with the Simpson's rule using the table below:

xx0022446688
f(x)f{\left(x \right)}−1-155002277

Solution

The Simpson's 1/3 rule approximates the integral using parabolas: ∫abf(x) dx≈∑i=1n−12Δxi3(f(x2i−1)+4f(x2i)+f(x2i+1))\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 nn is the number of points and Δxi\Delta x_{i} is the length of subinterval no. 2i−12 i - 1.

∫08f(x) dx≈2−03(f(0)+4f(2)+f(4))+6−43(f(4)+4f(6)+f(8))\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, ∫08f(x) dx≈2−03(−1+(4)⋅(5)+0)+6−43(0+(4)⋅(2)+7)=683.\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

∫08f(x) dx≈683≈22.666666666666667\int\limits_{0}^{8} f{\left(x \right)}\, dx\approx \frac{68}{3}\approx 22.666666666666667A