Approximate $$$\int\limits_{0}^{\pi} \sin{\left(x^{2} \right)}\, dx$$$ with $$$n = 5$$$ using the midpoint rule

Approximate the integral $$$\int\limits_{0}^{\pi} \sin{\left(x^{2} \right)}\, dx$$$ with $$$n = 5$$$ using the midpoint rule.

The midpoint rule (also known as the midpoint approximation) uses the midpoint of a subinterval for computing the height of the approximating rectangle:

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

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

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

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

Divide the interval $$$\left[0, \pi\right]$$$ into $$$n = 5$$$ subintervals of the length $$$\Delta x = \frac{\pi}{5}$$$ with the following endpoints: $$$a = 0$$$, $$$\frac{\pi}{5}$$$, $$$\frac{2 \pi}{5}$$$, $$$\frac{3 \pi}{5}$$$, $$$\frac{4 \pi}{5}$$$, $$$\pi = b$$$.

Now, just evaluate the function at the midpoints of the subintervals.

$$$f{\left(\frac{x_{0} + x_{1}}{2} \right)} = f{\left(\frac{0 + \frac{\pi}{5}}{2} \right)} = f{\left(\frac{\pi}{10} \right)} = \sin{\left(\frac{\pi^{2}}{100} \right)}\approx 0.098535890500574$$$

$$$f{\left(\frac{x_{1} + x_{2}}{2} \right)} = f{\left(\frac{\frac{\pi}{5} + \frac{2 \pi}{5}}{2} \right)} = f{\left(\frac{3 \pi}{10} \right)} = \sin{\left(\frac{9 \pi^{2}}{100} \right)}\approx 0.775978167711571$$$

$$$f{\left(\frac{x_{2} + x_{3}}{2} \right)} = f{\left(\frac{\frac{2 \pi}{5} + \frac{3 \pi}{5}}{2} \right)} = f{\left(\frac{\pi}{2} \right)} = \sin{\left(\frac{\pi^{2}}{4} \right)}\approx 0.624265952639699$$$

$$$f{\left(\frac{x_{3} + x_{4}}{2} \right)} = f{\left(\frac{\frac{3 \pi}{5} + \frac{4 \pi}{5}}{2} \right)} = f{\left(\frac{7 \pi}{10} \right)} = \sin{\left(\frac{49 \pi^{2}}{100} \right)}\approx -0.992356786508554$$$

$$$f{\left(\frac{x_{4} + x_{5}}{2} \right)} = f{\left(\frac{\frac{4 \pi}{5} + \pi}{2} \right)} = f{\left(\frac{9 \pi}{10} \right)} = \sin{\left(\frac{81 \pi^{2}}{100} \right)}\approx 0.990160389295942$$$

Finally, just sum up the above values and multiply by $$$\Delta x = \frac{\pi}{5}$$$: $$$\frac{\pi}{5} \left(0.098535890500574 + 0.775978167711571 + 0.624265952639699 - 0.992356786508554 + 0.990160389295942\right) = 0.940331217218375.$$$

$$$\int\limits_{0}^{\pi} \sin{\left(x^{2} \right)}\, dx\approx 0.940331217218375$$$A