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

The calculator will approximate the integral of $$$\sin{\left(x^{2} \right)}$$$ from $$$0$$$ to $$$\pi$$$ with $$$n = 5$$$ subintervals using the right endpoint approximation, with steps shown.

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


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

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

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 right endpoints of the subintervals.

$$$f{\left(x_{1} \right)} = f{\left(\frac{\pi}{5} \right)} = \sin{\left(\frac{\pi^{2}}{25} \right)}\approx 0.384608975077325$$$

$$$f{\left(x_{2} \right)} = f{\left(\frac{2 \pi}{5} \right)} = \sin{\left(\frac{4 \pi^{2}}{25} \right)}\approx 0.999965219254203$$$

$$$f{\left(x_{3} \right)} = f{\left(\frac{3 \pi}{5} \right)} = \sin{\left(\frac{9 \pi^{2}}{25} \right)}\approx -0.399952418334562$$$

$$$f{\left(x_{4} \right)} = f{\left(\frac{4 \pi}{5} \right)} = \sin{\left(\frac{16 \pi^{2}}{25} \right)}\approx 0.033355321355804$$$

$$$f{\left(x_{5} \right)} = f{\left(\pi \right)} = \sin{\left(\pi^{2} \right)}\approx -0.430301217000092$$$

Finally, just sum up the above values and multiply by $$$\Delta x = \frac{\pi}{5}$$$: $$$\frac{\pi}{5} \left(0.384608975077325 + 0.999965219254203 - 0.399952418334562 + 0.033355321355804 - 0.430301217000092\right) = 0.369247645681578.$$$


$$$\int\limits_{0}^{\pi} \sin{\left(x^{2} \right)}\, dx\approx 0.369247645681578$$$A