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
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
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)
Verified step by step guidance1
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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Video duration:
3mKey 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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Guided course
05:43
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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06:11Limits 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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Related Practice
Textbook Question
Solve the differential equation in Exercises 9–22.
21. (1/x)(dy/dx) = ye^(x²) + 2√y e^(x²)
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
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
37. lim (y → 0) (√(5y + 25) - 5) / y
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Textbook Question
In Exercises 9 and 10, use implicit differentiation to find dy/dx.
9. y^e^x = x^y + 1
Textbook Question
20. Solid of revolution The region between the curve y=1/(2√x) and the x-axis from x=1/4 to x=4 is revolved about the x-axis to generate a solid.
b. Find the centroid of the region.
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