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

The calculator will approximate the integral of $\sin{\left(x^{3} \right)}$ from $0$ to $3$ with $n = 6$ subintervals using the Riemann sum, with steps shown.

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

### Solution

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