# Riemann Sum Calculator for a Function

## Approximate an integral (given by a function) using the Riemann sum step by step

The calculator will approximate the definite integral using the Riemann sum and the sample points of your choice: left endpoints, right endpoints, midpoints, or trapezoids.

Related calculator: Riemann Sum Calculator for a Table

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Approximate the integral $\int\limits_{0}^{2} \sqrt[3]{x^{4} + 1}\, dx$ with $n = 4$ using the left Riemann sum.

### Solution

The left Riemann sum (also known as the left endpoint approximation) uses the left endpoint of a subinterval for computing the height of the approximating rectangle:

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

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

We have that $f{\left(x \right)} = \sqrt[3]{x^{4} + 1}$, $a = 0$, $b = 2$, and $n = 4$.

Therefore, $\Delta x = \frac{2 - 0}{4} = \frac{1}{2}$.

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

Now, just evaluate the function at the left endpoints of the subintervals.

$f{\left(x_{0} \right)} = f{\left(0 \right)} = 1$

$f{\left(x_{1} \right)} = f{\left(\frac{1}{2} \right)} = \frac{\sqrt[3]{17} \cdot 2^{\frac{2}{3}}}{4}\approx 1.020413775479337$

$f{\left(x_{2} \right)} = f{\left(1 \right)} = \sqrt[3]{2}\approx 1.259921049894873$

$f{\left(x_{3} \right)} = f{\left(\frac{3}{2} \right)} = \frac{2^{\frac{2}{3}} \sqrt[3]{97}}{4}\approx 1.82340825744217$

Finally, just sum up the above values and multiply by $\Delta x = \frac{1}{2}$: $\frac{1}{2} \left(1 + 1.020413775479337 + 1.259921049894873 + 1.82340825744217\right) = 2.55187154140819.$

$\int\limits_{0}^{2} \sqrt[3]{x^{4} + 1}\, dx\approx 2.55187154140819$A