Approximate $$$\int\limits_{0}^{3} \sin{\left(x^{3} \right)}\, dx$$$ with $$$n = 6$$$ using the Riemann sum

Approximate the integral $$$\int\limits_{0}^{3} \sin{\left(x^{3} \right)}\, dx$$$ with $$$n = 6$$$ 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 $$$f{\left(x \right)} = \sin{\left(x^{3} \right)}$$$, $$$a = 0$$$, $$$b = 3$$$, and $$$n = 6$$$.

Therefore, $$$\Delta x = \frac{3 - 0}{6} = \frac{1}{2}$$$.

Divide the interval $$$\left[0, 3\right]$$$ into $$$n = 6$$$ subintervals of the length $$$\Delta x = \frac{1}{2}$$$ with the following endpoints: $$$a = 0$$$, $$$\frac{1}{2}$$$, $$$1$$$, $$$\frac{3}{2}$$$, $$$2$$$, $$$\frac{5}{2}$$$, $$$3 = b$$$.

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

$$$f{\left(x_{0} \right)} = f{\left(0 \right)} = 0$$$

$$$f{\left(x_{1} \right)} = f{\left(\frac{1}{2} \right)} = \sin{\left(\frac{1}{8} \right)}\approx 0.124674733385228$$$

$$$f{\left(x_{2} \right)} = f{\left(1 \right)} = \sin{\left(1 \right)}\approx 0.841470984807897$$$

$$$f{\left(x_{3} \right)} = f{\left(\frac{3}{2} \right)} = \sin{\left(\frac{27}{8} \right)}\approx -0.231293812402022$$$

$$$f{\left(x_{4} \right)} = f{\left(2 \right)} = \sin{\left(8 \right)}\approx 0.989358246623382$$$

$$$f{\left(x_{5} \right)} = f{\left(\frac{5}{2} \right)} = \sin{\left(\frac{125}{8} \right)}\approx 0.082868129330598$$$

Finally, just sum up the above values and multiply by $$$\Delta x = \frac{1}{2}$$$: $$$\frac{1}{2} \left(0 + 0.124674733385228 + 0.841470984807897 - 0.231293812402022 + 0.989358246623382 + 0.082868129330598\right) = 0.903539140872542.$$$

$$$\int\limits_{0}^{3} \sin{\left(x^{3} \right)}\, dx\approx 0.903539140872542$$$A