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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 7.3.147
Chapter 7, Problem 7.3.147

147. Find the area of the region between the curve y = 2x / (1 + x²) and the interval −2 ≤ x ≤ 2 of the x-axis.

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Identify the function and the interval: The function given is \(y = \frac{2x}{1 + x^{2}}\) and the interval is \(-2 \leq x \leq 2\).
Determine the points where the curve intersects the x-axis within the interval by setting \(y = 0\): Solve \(\frac{2x}{1 + x^{2}} = 0\) to find the roots.
Split the integral at the points where the function crosses the x-axis to handle areas above and below the x-axis separately, ensuring the total area is positive.
Set up the definite integral(s) for the area between the curve and the x-axis: Calculate \(\int |y| \, dx\) over the appropriate subintervals, which means integrating \(\left| \frac{2x}{1 + x^{2}} \right|\) from \(-2\) to \(2\).
Evaluate the integral(s) by integrating \(\frac{2x}{1 + x^{2}}\) or its negative depending on the sign of \(y\) in each subinterval, then sum the results to find the total area.

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Key Concepts

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

Definite Integral for Area Calculation

The definite integral of a function over an interval gives the net area between the curve and the x-axis. To find the total area, especially when the function crosses the x-axis, one must consider the absolute value of the function or split the integral at points where the function changes sign.
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Definition of the Definite Integral

Finding Points of Intersection with the x-axis

To determine where the curve crosses the x-axis, set y = 0 and solve for x. These points divide the interval into subintervals where the function is either positive or negative, which is essential for correctly calculating the total area.
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Finding Area Between Curves that Cross on the Interval Example 3

Integration of Rational Functions

The given function is a rational function of the form 2x/(1 + x²). Integrating such functions often involves substitution methods, such as letting u = 1 + x², to simplify the integral and find an antiderivative.
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Intro to Rational Functions
Related Practice
Textbook Question

Each of Exercises 25–36 gives a formula for a function y=f(x). In each case, find f^(-1)(x) and identify the domain and range of f^(-1). As a check, show that f(f^(-1)(x))=f^(-1)(f(x))=x.

f(x) = x⁵

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