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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Your Input

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 $$$\Delta x = \frac{4}{5}$$$: $$$\frac{4}{5} \left(1.732050807568877 + 1.495196773630485 + 1.414213819387789 + 1.515144715776502 + 1.730085700215823\right) = 6.309353453263581.$$$

Answer

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