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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 7.1.71a
Chapter 7, Problem 7.1.71a

Zero net area Consider the function f(x) = (1 − x)/x
a. Are there numbers 0 < a < 1 such that ∫₁₋ₐ¹⁺ᵃ f(x) dx = 0?

Verified step by step guidance
1
First, understand the problem: we want to find if there exists a number \( a \) with \( 0 < a < 1 \) such that the definite integral \( \int_{1 - a}^{1 + a} f(x) \, dx = 0 \), where \( f(x) = \frac{1 - x}{x} \).
Rewrite the function \( f(x) \) to simplify integration: \( f(x) = \frac{1}{x} - 1 \). This allows us to split the integral into two simpler integrals.
Set up the integral as \( \int_{1 - a}^{1 + a} \left( \frac{1}{x} - 1 \right) dx = \int_{1 - a}^{1 + a} \frac{1}{x} \, dx - \int_{1 - a}^{1 + a} 1 \, dx \).
Evaluate each integral separately: \( \int \frac{1}{x} \, dx = \ln|x| \) and \( \int 1 \, dx = x \). Substitute the limits \( 1 - a \) and \( 1 + a \) into each result.
Combine the evaluated integrals to form the expression \( \left[ \ln|x| \right]_{1 - a}^{1 + a} - \left[ x \right]_{1 - a}^{1 + a} \). Set this equal to zero and solve the resulting equation for \( a \) in the interval \( (0,1) \).

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Key Concepts

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

Definite Integral and Net Area

The definite integral of a function over an interval represents the net area between the function's graph and the x-axis. Positive areas above the axis add to the integral, while areas below subtract. Understanding how these areas combine is essential to determine when the integral equals zero.
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Definition of the Definite Integral

Behavior of Rational Functions Near Singularities

The function f(x) = (1 - x)/x has a singularity at x = 0, where it is undefined. Analyzing the function's behavior near this point and within the interval around 1 is crucial to evaluate the integral and understand how the function's sign changes affect the net area.
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Intro to Rational Functions

Symmetry and Interval Selection in Integration

Choosing symmetric intervals around a point (here, around x=1) can lead to cancellation of positive and negative areas in the integral. Recognizing how the function behaves on [1 - a, 1 + a] helps determine if there exists an 'a' such that the integral sums to zero.
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Improper Integrals: Infinite Intervals
Related Practice
Textbook Question

61–62. Points of intersection and area

a. Sketch the graphs of the functions f and g and find the x-coordinate of the points at which they intersect.


f(x) = sech x, g(x) = tanh x; the region bounded by the graphs of f, g, and the y-axis

Textbook Question

Visual approximation


a. Use a graphing utility to sketch the graph of y = coth x and then explain why ∫₅¹⁰ coth x dx ≈ 5.

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Textbook Question

Velocity of falling body Refer to Exercise 95, which gives the position function for a falling body. Use m = 75 kg and k = 0.2.


a. Confirm that the BASE jumper’s velocity t seconds after jumping is v(t) = d'(t) = √(mg/k) tanh (√(kg/m) t).

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Textbook Question

ln x is unbounded Use the following argument to show that lim (x → ∞) ln x = ∞ and lim (x → 0⁺) ln x = −∞.

a. Make a sketch of the function f(x) = 1/x on the interval [1, 2]. Explain why the area of the region bounded by y = f(x) and the x-axis on [1, 2] is ln 2.

Textbook Question

Evaluating hyperbolic functions Use a calculator to evaluate each expression or state that the value does not exist. Report answers accurate to four decimal places to the right of the decimal point.

a. coth 4

Textbook Question

Energy consumption On the first day of the year (t=0), a city uses electricity at a rate of 2000 MW. That rate is projected to increase at a rate of 1.3% per year.


a. Based on these figures, find an exponential growth function for the power (rate of electricity use) for the city.