Use the Midpoint Rule with n = 10 to approximate the integral.

Question

Quizplus · Accepted Answer

The Midpoint Rule for approximating an integral is:

$ \int_{a}^{b} f(x) dx \approx \frac{b-a}{n} \sum_{i=1}^{n} f(\frac{x_{i-1}+x_i}{2}) $

where $x_0 = a$ and $x_n = b$.

In this case, $a=0$, $b=1$, and $n=10$. So we have:

$ \int_{0}^{1} e^x dx \approx \frac{1-0}{10} \sum_{i=1}^{10} e^{\frac{i-1/2}{10}} $

$ \approx 0.1 \cdot (e^{0.05} + e^{0.15} + e^{0.25} + \dots + e^{0.95}) $

Using a calculator, we get:

$ \approx 0.1 \cdot (1.05127 + 1.16183 + 1.28403 + \dots + 2.4596) $

$ \approx 0.1 \cdot 14.9143 \approx 1.491 $

Out of the answer choices, only (B) is closest to this value.

Use the Midpoint Rule with N = 10 to Approximate