교과서 질문
In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.
∫ from 2 to ∞ of ((x dx) / √(x⁴ - 1))
Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
8장, 문제 8.8.87
87. Find the area of the region that lies between the curves y = sec x and y = tan x from x = 0 to x = π/2.
검증된 단계별 안내1
Identify the curves and the interval: We are given two functions, \(y = \sec x\) and \(y = \tan x\), and we want to find the area between them from \(x = 0\) to \(x = \frac{\pi}{2}\).
Determine which function is on top and which is on the bottom in the interval \([0, \frac{\pi}{2})\): Since \(\sec x = \frac{1}{\cos x}\) and \(\tan x = \frac{\sin x}{\cos x}\), and for \(x\) in this interval \(\sin x\) is between 0 and 1, it follows that \(\sec x \geq \tan x\). So, the area between the curves is given by the integral of \((\sec x - \tan x)\) over \([0, \frac{\pi}{2})\).
Set up the definite integral for the area: The area \(A\) is given by
\(A = \int_0^{\frac{\pi}{2}} (\sec x - \tan x) \, dx\).
Recall the antiderivatives: The integral of \(\sec x\) is \(\ln |\sec x + \tan x| + C\), and the integral of \(\tan x\) is \(-\ln |\cos x| + C\). Use these to find the antiderivative of the integrand.
Evaluate the definite integral by substituting the limits \(x = 0\) and \(x = \frac{\pi}{2}\) into the antiderivative expression, and then compute the difference to find the area.
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3m주요 개념
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Definite Integrals for Area Between Curves
The area between two curves over an interval is found by integrating the difference of their functions. Specifically, the integral of the upper function minus the lower function from the lower to the upper limit gives the enclosed area.
추천 영상:
05:23
Finding Area Between Curves on a Given Interval
Properties of Trigonometric Functions sec x and tan x
Understanding the behavior of sec x and tan x on the interval [0, π/2) is crucial. Both functions increase and approach infinity as x approaches π/2, with sec x always greater than tan x in this interval, which determines the order of subtraction in the integral.
추천 영상:
가이드 코스
06:21Properties of Functions
Integration Techniques for Trigonometric Functions
Evaluating the integral of sec x and tan x requires knowledge of their antiderivatives. The integral of sec x is ln|sec x + tan x|, and the integral of tan x is -ln|cos x|, which are essential for computing the exact area.
추천 영상:
가이드 코스
6:04Introduction to Trigonometric Functions
관련 실천
교과서 질문
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교과서 질문
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교과서 질문
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교과서 질문
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교과서 질문
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