# Simpson's Rule Calculator for a Function

An online calculator for approximating a definite integral using Simpson's (parabolic) 1/3 rule, with steps shown.

Related calculator: Simpson's Rule Calculator for a Table

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Approximate the integral $\int\limits_{0}^{1} \frac{1}{\sqrt[3]{x^{5} + 7}}\, dx$ with $n = 4$ using the Simpson's rule.

## Solution

The Simpson's 1/3 rule (also known as the parabolic rule) uses parabolas to approximate the area:

$\int\limits_{a}^{b} f{\left(x \right)}\, dx\approx \frac{\Delta x}{3} \left(f{\left(x_{0} \right)} + 4 f{\left(x_{1} \right)} + 2 f{\left(x_{2} \right)} + 4 f{\left(x_{3} \right)} + 2 f{\left(x_{4} \right)}+\dots+4 f{\left(x_{n-3} \right)} + 2 f{\left(x_{n-2} \right)} + 4 f{\left(x_{n-1} \right)} + f{\left(x_{n} \right)}\right)$

where $\Delta x = \frac{b - a}{n}$.

We have that $f{\left(x \right)} = \frac{1}{\sqrt[3]{x^{5} + 7}}$, $a = 0$, $b = 1$, and $n = 4$.

Therefore, $\Delta x = \frac{1 - 0}{4} = \frac{1}{4}$.

Divide the interval $\left[0, 1\right]$ into $n = 4$ subintervals of the length $\Delta x = \frac{1}{4}$ with the following endpoints: $a = 0$, $\frac{1}{4}$, $\frac{1}{2}$, $\frac{3}{4}$, $1 = b$.

Now, just evaluate the function at these endpoints.

$f{\left(x_{0} \right)} = f{\left(0 \right)} = \frac{7^{\frac{2}{3}}}{7}\approx 0.52275795857471$

$4 f{\left(x_{1} \right)} = 4 f{\left(\frac{1}{4} \right)} = \frac{32 \sqrt[3]{2} \cdot 7169^{\frac{2}{3}}}{7169}\approx 2.09093460413808$

$2 f{\left(x_{2} \right)} = 2 f{\left(\frac{1}{2} \right)} = \frac{4 \sqrt[3]{15} \cdot 2^{\frac{2}{3}}}{15}\approx 1.043964704311697$

$4 f{\left(x_{3} \right)} = 4 f{\left(\frac{3}{4} \right)} = \frac{32 \sqrt[3]{2} \cdot 7411^{\frac{2}{3}}}{7411}\approx 2.067923042238355$

$f{\left(x_{4} \right)} = f{\left(1 \right)} = \frac{1}{2} = 0.5$

Finally, just sum up the above values and multiply by $\frac{\Delta x}{3} = \frac{1}{12}$: $\frac{1}{12} \left(0.52275795857471 + 2.09093460413808 + 1.043964704311697 + 2.067923042238355 + 0.5\right) = 0.518798359105237.$

$\int\limits_{0}^{1} \frac{1}{\sqrt[3]{x^{5} + 7}}\, dx\approx 0.518798359105237$A