Find the area of the region bounded above by $$y=e^x$$​, bounded below by $$y=x$$​, and bounded on the sides by $$x=0 \_0$$​ and $$x=1$$​.

Correct option is A

​$$\begin{aligned}&\bullet \ \text{Upper boundary: } y = e^x \\&\bullet \ \text{Lower boundary: } y = x \\&\bullet \ \text{Left boundary: } x = 0 \\&\bullet \ \text{Right boundary: } x = 1 \\\\&\textbf{Set Up the Integral} \\&\text{The area } A \text{ between the curves from } x = 0 \text{ to } x = 1 \text{ is given by the integral of the upper function minus} \\&\text{the lower function:} \\&\qquad A = \int_{0}^{1} (e^x - x) dx \\\\&\textbf{Compute the Integral} \\&\text{Evaluate the integral step by step:} \\&\qquad A = \int_{0}^{1} e^x dx - \int_{0}^{1} x dx \\&\qquad A = \left[ e^x \right]_{0}^{1} - \left[ \frac{x^2}{2} \right]_{0}^{1} \\&\qquad A = (e^1 - e^0) - \left( \frac{1^2}{2} - \frac{0^2}{2} \right) \\&\qquad A = (e - 1) - \left( \frac{1}{2} - 0 \right) \\&\qquad A = e - 1 - \frac{1}{2} \\&\qquad A = e - \frac{3}{2}\end{aligned}$$​​