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

The calculator will approximate the integral of $30 \sin{\left(2 x \right)}$ from $1$ to $2$ with $n = 8$ subintervals using the Riemann sum, with steps shown.

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Approximate the integral $\int\limits_{1}^{2} 30 \sin{\left(2 x \right)}\, dx$ with $n = 8$ 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 $f{\left(x \right)} = 30 \sin{\left(2 x \right)}$, $a = 1$, $b = 2$, 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