7. Let A(t) be the area of the region in the first quadrant enclosed by the coordinate axes, the curve y=e^(-x), and the vertical line x=t, t>0. Let V(t) be the volume of the solid generated by revolving the region about the x-axis. Find the following limits.
a. lim(x→∞)A(t)

Verified step by step guidance

1. First, understand the region described: it is bounded by the x-axis (y=0), the y-axis (x=0), the curve \(y = e^{-x}\), and the vertical line \(x = t\) where \(t > 0\).
2. To find the area \(A(t)\) of this region, set up the definite integral of the function \(y = e^{-x}\) from \(x=0\) to \(x=t\): \[A(t) = \int_0^t e^{-x} \, dx\]
3. Evaluate the integral symbolically (do not compute the final value yet): \[\int e^{-x} \, dx = -e^{-x} + C\] So, \[A(t) = [-e^{-x}]_0^t = (-e^{-t}) - (-e^{0}) = 1 - e^{-t}\]
4. Now, to find the limit as \(t\) approaches infinity, consider the behavior of \(e^{-t}\) as \(t \to \infty\). Since \(e^{-t}\) approaches 0, the expression \(1 - e^{-t}\) approaches 1.
5. Therefore, the limit is \[\lim_{t \to \infty} A(t) = 1\] This represents the total area under the curve \(y = e^{-x}\) from 0 to infinity in the first quadrant.

Key Concepts

Definite Integral as Area Under a Curve

The area A(t) under the curve y = e^(-x) from x = 0 to x = t is found using a definite integral. This integral sums the infinitesimal areas of vertical slices, representing the total area enclosed by the curve, axes, and vertical line at x = t.

Limit of a Function as t Approaches Infinity

Evaluating lim(t→∞) A(t) involves understanding the behavior of the integral as the upper limit grows without bound. This often requires recognizing improper integrals and determining if the area converges to a finite value or diverges.

Exponential Decay Function

The function y = e^(-x) is an exponential decay curve that approaches zero as x increases. Its properties ensure the area under the curve from 0 to infinity converges, which is key to finding the limit of A(t) as t approaches infinity.