# Approximate $\int\limits_{1}^{3} \frac{\sin{\left(x \right)}}{x}\, dx$ with $n = 2$ using the left endpoint approximation

The calculator will approximate the integral of $\frac{\sin{\left(x \right)}}{x}$ from $1$ to $3$ with $n = 2$ subintervals using the left endpoint approximation, with steps shown.

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Approximate the integral $\int\limits_{1}^{3} \frac{\sin{\left(x \right)}}{x}\, dx$ with $n = 2$ using the left endpoint approximation.

### 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 $f{\left(x \right)} = \frac{\sin{\left(x \right)}}{x}$, $a = 1$, $b = 3$, and $n = 2$.

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

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

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

$f{\left(x_{0} \right)} = f{\left(1 \right)} = \sin{\left(1 \right)}\approx 0.841470984807897$

$f{\left(x_{1} \right)} = f{\left(2 \right)} = \frac{\sin{\left(2 \right)}}{2}\approx 0.454648713412841$

Finally, just sum up the above values and multiply by $\Delta x = 1$: $1 \left(0.841470984807897 + 0.454648713412841\right) = 1.296119698220738.$

$\int\limits_{1}^{3} \frac{\sin{\left(x \right)}}{x}\, dx\approx 1.296119698220738$A