Approximate $$$\int\limits_{1}^{7} 14 e^{x^{2} - 1}\, dx$$$ with $$$n = 3$$$ using the left endpoint approximation

Approximate the integral $$$\int\limits_{1}^{7} 14 e^{x^{2} - 1}\, dx$$$ with $$$n = 3$$$ using the left endpoint approximation.

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 $$$f{\left(x \right)} = 14 e^{x^{2} - 1}$$$, $$$a = 1$$$, $$$b = 7$$$, and $$$n = 3$$$.

Therefore, $$$\Delta x = \frac{7 - 1}{3} = 2$$$.

Divide the interval $$$\left[1, 7\right]$$$ into $$$n = 3$$$ subintervals of the length $$$\Delta x = 2$$$ with the following endpoints: $$$a = 1$$$, $$$3$$$, $$$5$$$, $$$7 = b$$$.

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

$$$f{\left(x_{0} \right)} = f{\left(1 \right)} = 14$$$

$$$f{\left(x_{1} \right)} = f{\left(3 \right)} = 14 e^{8}\approx 41733.411818584195846$$$

$$$f{\left(x_{2} \right)} = f{\left(5 \right)} = 14 e^{24}\approx 3.7084770981780861211794827 \cdot 10^{11}$$$

Finally, just sum up the above values and multiply by $$$\Delta x = 2$$$: $$$2 \left(14 + 41733.411818584195846 + 3.7084770981780861211794827 \cdot 10^{11}\right) = 7.41695503130440861404288232 \cdot 10^{11}.$$$

$$$\int\limits_{1}^{7} 14 e^{x^{2} - 1}\, dx\approx 7.41695503130440861404288232 \cdot 10^{11}$$$A