# 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.

### Solution

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