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

The calculator will approximate the integral of $14 e^{x^{2} - 1}$ from $1$ to $7$ with $n = 3$ subintervals using the left endpoint approximation, with steps shown.

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Approximate the integral $\int\limits_{1}^{7} 14 e^{x^{2} - 1}\, dx$ with $n = 3$ 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 $f{\left(x \right)} = 14 e^{x^{2} - 1}$, $a = 1$, $b = 7$, 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