Ch. 8 - Integration Techniques

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 8 - Integration Techniques

Problema 8.R.123

Capitolo 8, Problema 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].

Guida verificata passo dopo passo

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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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.

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05:23

Finding Area Between Curves on a Given Interval

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.

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Percorso guidato

06:21

Properties of Functions

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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Percorso guidato

6:04

Introduction to Trigonometric Functions

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