Approximate $$$\int\limits_{1}^{2} 30 \sin{\left(2 x \right)}\, dx$$$ with $$$n = 8$$$ using the Riemann sum

Approximate the integral $$$\int\limits_{1}^{2} 30 \sin{\left(2 x \right)}\, dx$$$ with $$$n = 8$$$ using the left Riemann sum.

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

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

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

We have that $$$f{\left(x \right)} = 30 \sin{\left(2 x \right)}$$$, $$$a = 1$$$, $$$b = 2$$$, and $$$n = 8$$$.

Therefore, $$$\Delta x = \frac{2 - 1}{8} = \frac{1}{8}$$$.

Divide the interval $$$\left[1, 2\right]$$$ into $$$n = 8$$$ subintervals of the length $$$\Delta x = \frac{1}{8}$$$ with the following endpoints: $$$a = 1$$$, $$$\frac{9}{8}$$$, $$$\frac{5}{4}$$$, $$$\frac{11}{8}$$$, $$$\frac{3}{2}$$$, $$$\frac{13}{8}$$$, $$$\frac{7}{4}$$$, $$$\frac{15}{8}$$$, $$$2 = b$$$.

Now, just evaluate the function at the left endpoints of the subintervals.

$$$f{\left(x_{0} \right)} = f{\left(1 \right)} = 30 \sin{\left(2 \right)}\approx 27.278922804770451$$$

$$$f{\left(x_{1} \right)} = f{\left(\frac{9}{8} \right)} = 30 \sin{\left(\frac{9}{4} \right)}\approx 23.342195906637637$$$

$$$f{\left(x_{2} \right)} = f{\left(\frac{5}{4} \right)} = 30 \sin{\left(\frac{5}{2} \right)}\approx 17.954164323118695$$$

$$$f{\left(x_{3} \right)} = f{\left(\frac{11}{8} \right)} = 30 \sin{\left(\frac{11}{4} \right)}\approx 11.449829761569951$$$

$$$f{\left(x_{4} \right)} = f{\left(\frac{3}{2} \right)} = 30 \sin{\left(3 \right)}\approx 4.233600241796017$$$

$$$f{\left(x_{5} \right)} = f{\left(\frac{13}{8} \right)} = 30 \sin{\left(\frac{13}{4} \right)}\approx -3.245854035903251$$$

$$$f{\left(x_{6} \right)} = f{\left(\frac{7}{4} \right)} = 30 \sin{\left(\frac{7}{2} \right)}\approx -10.523496830688595$$$

$$$f{\left(x_{7} \right)} = f{\left(\frac{15}{8} \right)} = 30 \sin{\left(\frac{15}{4} \right)}\approx -17.146839562270313$$$

Finally, just sum up the above values and multiply by $$$\Delta x = \frac{1}{8}$$$: $$$\frac{1}{8} \left(27.278922804770451 + 23.342195906637637 + 17.954164323118695 + 11.449829761569951 + 4.233600241796017 - 3.245854035903251 - 10.523496830688595 - 17.146839562270313\right) = 6.667815326128824.$$$

$$$\int\limits_{1}^{2} 30 \sin{\left(2 x \right)}\, dx\approx 6.667815326128824$$$A