Definition of the Definite Integral — Question 5

Question 5- Solution

Partition the interval $[0,1]$ into $n$ equal subintervals. Each subinterval has width $$\Delta x=\frac{1}{n}.$$

Using right endpoints, the $i$th sample point is $$x_i=i\Delta x=\frac{i}{n}.$$

Evaluate the function at each sample point: $$f(x_i)=e^{i/n}.$$

The Riemann sum is $$\sum_{i=1}^{n} f(x_i)\,\Delta x = \frac{1}{n}\sum_{i=1}^{n} e^{i/n}.$$

Thus, $$\int_{0}^{1} e^{x}\,dx = \lim_{n\to\infty} \frac{1}{n}\sum_{i=1}^{n} e^{i/n}.$$

Evaluating the definite integral, $$\int_{0}^{1} e^{x}\,dx = \bigl[e^{x}\bigr]_{0}^{1} = e-1.$$

$$\boxed{e-1}$$