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Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 7 - Transcendental FunctionsProblem 7.AAE.7a
Chapter 7, Problem 7.AAE.7a

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)

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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 \(t > 0\).
To find the area \(A(t)\) of this region, set up the definite integral of the function \(y = e^{-x}\) from \(x=0\) to \(x=t\): \[A(t) = \int_0^t e^{-x} \, dx\]
Evaluate the integral symbolically (do not compute the final value yet): \[\int e^{-x} \, dx = -e^{-x} + C\] So, \[A(t) = [-e^{-x}]_0^t = (-e^{-t}) - (-e^{0}) = 1 - e^{-t}\]
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.
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.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
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Definition of the Definite Integral

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.
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Limits of Rational Functions: Denominator = 0

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.
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Exponential Growth & Decay
Related Practice
Textbook Question

17. Even-odd decompositions

b. If f(x) = f_E(x) + f_O(x) is the sum of an even function f_E(x) and an odd function f_O(x), then show that

f_E(x) = (f(x)+f(-x))/2 and f_O(x) = (f(x)-f(-x))/2

Textbook Question

13. For what x>0 does x^(x^x) = (x^x)^x? Give reasons for your answer.

Textbook Question

15. Find f'(2) if f(x) = e^(g(x)) and g(x) = ∫(from 2 to x) t/(1+t⁴)dt.

Textbook Question

Find the areas between the curves y=2(log_2(x))/x and y=2(log_4(x))/x and the x-axis from x=1 to x=e. What is the ratio of the larger area to the smaller?

Textbook Question

Find the limits in Exercises 1–6.

5. lim(n→∞) (1/(n+1) + 1/(n+2) + ... + 1/(2n))

Textbook Question

In Exercises 9 and 10, use implicit differentiation to find dy/dx.

9. y^e^x = x^y + 1