# Left Endpoint Approximation Calculator for a Function

An online calculator for approximating a definite integral using 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 $$a = 0$$$, $$b = 4$$$, $$n = 5$$$. 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. $$f{\left(x_{0} \right)} = f{\left(0 \right)} = \sqrt{3}\approx 1.73205080756888$$$

$$f{\left(x_{1} \right)} = f{\left(\frac{4}{5} \right)} = \sqrt{\cos^{4}{\left(\frac{4}{5} \right)} + 2}\approx 1.49519677363048$$$$$f{\left(x_{2} \right)} = f{\left(\frac{8}{5} \right)} = \sqrt{\cos^{4}{\left(\frac{8}{5} \right)} + 2}\approx 1.41421381938779$$$

$$f{\left(x_{3} \right)} = f{\left(\frac{12}{5} \right)} = \sqrt{\cos^{4}{\left(\frac{12}{5} \right)} + 2}\approx 1.5151447157765$$$$$f{\left(x_{4} \right)} = f{\left(\frac{16}{5} \right)} = \sqrt{\cos^{4}{\left(\frac{16}{5} \right)} + 2}\approx 1.73008570021582$$$

Finally, just sum up the above values and multiply by $$\Delta x = \frac{4}{5}$$$: $$\frac{4}{5} \left(1.73205080756888 + 1.49519677363048 + 1.41421381938779 + 1.5151447157765 + 1.73008570021582\right) = 6.30935345326358.$$$

$$\int\limits_{0}^{4} \sqrt{\cos^{4}{\left(x \right)} + 2}\, dx\approx 6.30935345326358$$\$A