Exercises 83–86 are about the infinite region in the first quadrant between the curve y = e^(-x) and the x-axis.
83. Find the area of the region.

Verified step by step guidance

1. Identify the region whose area you want to find. Here, it is the area in the first quadrant bounded by the curve \(y = e^{-x}\) and the x-axis.
2. Set up the definite integral to find the area. Since the region is in the first quadrant and bounded by the x-axis, the limits of integration for \(x\) are from 0 to \(\infty\). The area \(A\) can be expressed as \(A = \int_0^{\infty} e^{-x} \, dx\).
3. Recall the integral formula for the exponential function: \(\int e^{ax} \, dx = \frac{1}{a} e^{ax} + C\) for \(a \neq 0\). In this case, \(a = -1\).
4. Evaluate the improper integral by taking the limit as the upper bound approaches infinity: \(A = \lim_{t \to \infty} \int_0^t e^{-x} \, dx\).
5. Compute the definite integral using the antiderivative and then apply the limit to find the area.

Key Concepts

Definite Integral as Area Under a Curve

The definite integral of a function over an interval represents the net area between the curve and the x-axis within that interval. For positive functions, this corresponds to the actual area bounded by the curve, the x-axis, and the vertical limits.

Improper Integrals and Infinite Limits

When the region extends infinitely, such as from a finite point to infinity, the integral is called an improper integral. Evaluating it involves taking a limit as the upper bound approaches infinity to determine if the area converges to a finite value.

Exponential Decay Function y = e^(-x)

The function y = e^(-x) is an exponential decay curve that approaches zero as x approaches infinity. Its properties ensure that the area under the curve from zero to infinity converges, making it suitable for calculating finite areas over infinite intervals.