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Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblem 7.1.67e
Chapter 7, Problem 7.1.67e

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume x > 0 and y > 0.


e. The area under the curve y = 1/x and the x-axis on the interval [1, e] is 1.

Verified step by step guidance
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Recall that the area under the curve \( y = \frac{1}{x} \) from \( x = 1 \) to \( x = e \) is given by the definite integral \( \int_1^e \frac{1}{x} \, dx \).
Set up the integral: \( \int_1^e \frac{1}{x} \, dx \). This integral represents the area under the curve between the given limits.
Use the fact that the integral of \( \frac{1}{x} \) with respect to \( x \) is \( \ln|x| + C \). So, \( \int \frac{1}{x} \, dx = \ln|x| + C \).
Evaluate the definite integral by applying the Fundamental Theorem of Calculus: \( \int_1^e \frac{1}{x} \, dx = \left[ \ln x \right]_1^e = \ln e - \ln 1 \).
Since \( \ln e = 1 \) and \( \ln 1 = 0 \), the area under the curve on the interval \([1, e]\) is \( 1 - 0 = 1 \). Therefore, the statement is true.

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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 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 integral equals the actual area under the curve. Understanding this helps interpret the integral of y = 1/x from 1 to e as the area under that curve.
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Definition of the Definite Integral

Integral of 1/x and the Natural Logarithm

The integral of 1/x with respect to x is the natural logarithm function, ln|x| + C. This means that the definite integral from 1 to e of 1/x dx equals ln(e) - ln(1), which simplifies to 1 - 0 = 1. Recognizing this connection is key to evaluating the area under y = 1/x.
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Integrals of Natural Exponential Functions (e^x)

Properties of the Number e

The number e is the base of the natural logarithm, approximately 2.718, and satisfies ln(e) = 1. This property is essential when evaluating integrals involving 1/x, as it directly determines the value of the definite integral from 1 to e, confirming the area under the curve.
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The Number e
Related Practice
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Textbook Question

Terminal velocity Refer to Exercises 95 and 96.


d. How tall must a cliff be so that the BASE jumper (m = 75 kg and k = 0.2) reaches 95% of terminal velocity? Assume the jumper needs at least 300 m at the end of free fall to deploy the chute and land safely.

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


d. If the rate constant of an exponential growth function is increased, its doubling time is decreased.

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