91. [Use of Tech] Regions bounded by exponentials Let a > 0 and let R be the region bounded by the graph of y = e^(-a·x) and the x-axis
on the interval [b, ∞).
a. Find A(a,b), the area of R as a function of a and b.

Verified step by step guidance

1. Identify the region R bounded by the curve \(y = e^{-a \cdot x}\) and the x-axis on the interval \([b, \infty)\). Since the curve is above the x-axis for \(a > 0\), the area can be found by integrating the function from \(x = b\) to \(x = \infty\).
2. Set up the integral for the area \(A(a,b)\) as: \(A(a,b) = \int_{b}^{\infty} e^{-a \cdot x} \, dx\)
3. Recall the integral formula for the exponential function: \(\int e^{kx} \, dx = \frac{1}{k} e^{kx} + C\), where \(k\) is a constant. Here, \(k = -a\).
4. Evaluate the definite integral by applying the limits from \(b\) to \(\infty\): \(A(a,b) = \left[ \frac{e^{-a \cdot x}}{-a} \right]_{b}^{\infty}\)
5. Calculate the limit as \(x \to \infty\) of \(e^{-a \cdot x}\), which approaches 0 for \(a > 0\), and then substitute \(x = b\) to express the area \(A(a,b)\) in terms of \(a\) and \(b\).

Key Concepts

Definite Integrals and Area Under a Curve

The area under a curve between two points is found using definite integrals. For a function f(x), the area from x = b to x = c is the integral of f(x) dx over [b, c]. When the upper limit is infinity, improper integrals are used to evaluate the area.

Exponential Functions and Their Properties

Exponential functions like y = e^(-a·x) decay rapidly as x increases when a > 0. Understanding their behavior helps in setting up integrals and evaluating limits, especially for infinite intervals where the function approaches zero.

Improper Integrals and Convergence

When integrating over an infinite interval, the integral is called improper. To find the area, one must evaluate the limit of the integral as the upper bound approaches infinity and verify that this limit converges to a finite value.