Ch. 8 - Integration Techniques

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 8 - Integration Techniques

Problem 8.R.123

Chapter 8, Problem 8.R.123

123. Region between curves Find the area of the region bounded by the graphs of y = tan(x) and y = sec(x) on the interval [0, π/4].

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Identify the two functions given: \(y = \tan(x)\) and \(y = \sec(x)\), and the interval \([0, \frac{\pi}{4}]\) over which we want to find the area between the curves.

Determine which function is on top and which is on the bottom on the interval \([0, \frac{\pi}{4}]\). This is done by comparing \(\tan(x)\) and \(\sec(x)\) at points in the interval.

Set up the integral for the area between the curves as \(\int_0^{\frac{\pi}{4}} (\text{top function} - \text{bottom function}) \, dx\).

Write the integral explicitly using the functions identified in step 2, for example, \(\int_0^{\frac{\pi}{4}} (\sec(x) - \tan(x)) \, dx\) if \(\sec(x)\) is on top.

Evaluate the integral by finding the antiderivatives of \(\sec(x)\) and \(\tan(x)\) separately, then apply the Fundamental Theorem of Calculus to compute the definite integral over \([0, \frac{\pi}{4}]\).

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Key Concepts

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

Area Between Curves

The area between two curves y = f(x) and y = g(x) over an interval [a, b] is found by integrating the difference of the functions, |f(x) - g(x)|, with respect to x. This requires identifying which function is on top within the interval to set up the integral correctly.

Properties of Trigonometric Functions (tan(x) and sec(x))

Understanding the behavior and values of tan(x) and sec(x) on [0, π/4] is essential. Both functions are positive and increasing there, but sec(x) is always greater than or equal to tan(x) in this interval, which helps determine the correct order for subtraction in the integral.

Integration of Trigonometric Functions

Calculating the area requires integrating expressions involving tan(x) and sec(x). Familiarity with the integrals ∫tan(x) dx = -ln|cos(x)| + C and ∫sec(x) dx = ln|sec(x) + tan(x)| + C is crucial for evaluating the definite integral accurately.

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