# Left Endpoint Approximation Calculator for a Function

An online calculator for approximating the definite integral using the left endpoints (the left Riemann sum), with steps shown.

Related calculator: Left Endpoint Approximation Calculator for a Table

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Approximate the integral $\int\limits_{0}^{4} \sqrt{\cos^{4}{\left(x \right)} + 2}\, dx$ with $n = 5$ using the left endpoint approximation.

## Solution

The left Riemann sum (also known as the left endpoint approximation) uses the left endpoints of a subinterval:

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

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

We have that $f{\left(x \right)} = \sqrt{\cos^{4}{\left(x \right)} + 2}$, $a = 0$, $b = 4$, and $n = 5$.

Therefore, $\Delta x = \frac{4 - 0}{5} = \frac{4}{5}$.

Divide the interval $\left[0, 4\right]$ into $n = 5$ subintervals of the length $\Delta x = \frac{4}{5}$ with the following endpoints: $a = 0$, $\frac{4}{5}$, $\frac{8}{5}$, $\frac{12}{5}$, $\frac{16}{5}$, $4 = b$.

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

$f{\left(x_{0} \right)} = f{\left(0 \right)} = \sqrt{3}\approx 1.732050807568877$

$f{\left(x_{1} \right)} = f{\left(\frac{4}{5} \right)} = \sqrt{\cos^{4}{\left(\frac{4}{5} \right)} + 2}\approx 1.495196773630485$

$f{\left(x_{2} \right)} = f{\left(\frac{8}{5} \right)} = \sqrt{\cos^{4}{\left(\frac{8}{5} \right)} + 2}\approx 1.414213819387789$

$f{\left(x_{3} \right)} = f{\left(\frac{12}{5} \right)} = \sqrt{\cos^{4}{\left(\frac{12}{5} \right)} + 2}\approx 1.515144715776502$

$f{\left(x_{4} \right)} = f{\left(\frac{16}{5} \right)} = \sqrt{\cos^{4}{\left(\frac{16}{5} \right)} + 2}\approx 1.730085700215823$

Finally, just sum up the above values and multiply by $\Delta x = \frac{4}{5}$: $\frac{4}{5} \left(1.732050807568877 + 1.495196773630485 + 1.414213819387789 + 1.515144715776502 + 1.730085700215823\right) = 6.309353453263581.$

$\int\limits_{0}^{4} \sqrt{\cos^{4}{\left(x \right)} + 2}\, dx\approx 6.309353453263581$A