Question

Consider the region satisfying the inequalities. Find the area of the region. $$ y \leq e^{-x}, y \geq 0, x \geq 0 $$

Short answer

The area of the region is 1.

Step-by-step solution

Identify the Region

Plot the inequality \(y \leq e^{-x}\) and take note of the region defined. The graph will be the exponential decay function starting from y=1 when x=0 till y approaches 0 as x increases. The region will be the area under this curve in the first quadrant because the restrictions \(y \geq 0\) and \(x \geq 0\) specify positive values of y and x.

Formulate integral

Since the defined region is under a curve and above the x-axis in the first quadrant, the task is to integrate the function \(y = e^{-x}\) with respect to x. The bounds of the integration will be from 0 to \(\infty\) as the region goes indefinitely along the x-axis.

Solve the Integral

Finally, evaluate the integral \(\int_{0}^{\infty} e^{-x} dx\). This integral yields an area of unity (1) which is the area under the entire curve of \(e^{-x}\).