/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 93 $$\int_{7 \pi / 6}^{5 \pi / 4} \... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 93

$$\int_{7 \pi / 6}^{5 \pi / 4} \sec ^{2} x d x$$

Short Answer

Expert verified
The integral evaluates to \(1 - \frac{\sqrt{3}}{3}\).

Step by step solution

01

Identify the Integral Form

The integral we are dealing with is \( \int \sec^2 x \, dx \). The standard integral formula for \( \sec^2 x \) is its antiderivative, \( \tan x \).
02

Apply the Integral Formula

Using the antiderivative, we have: \( \int \sec^2 x \, dx = \tan x + C \). For a definite integral from \( a \) to \( b \), it becomes \( [\tan x]_a^b = \tan b - \tan a \).
03

Evaluate the Antiderivative at the Upper Limit

Evaluate \( \tan x \) at the upper limit: \( x = \frac{5\pi}{4} \). Since \( \tan \left( \frac{5\pi}{4} \right) = 1 \).
04

Evaluate the Antiderivative at the Lower Limit

Evaluate \( \tan x \) at the lower limit: \( x = \frac{7\pi}{6} \). Since \( \tan \left( \frac{7\pi}{6} \right) = \frac{1}{\sqrt{3}} \).
05

Subtract the Evaluated Values

The value of the definite integral is \( \tan \left( \frac{5\pi}{4} \right) - \tan \left( \frac{7\pi}{6} \right) = 1 - \frac{1}{\sqrt{3}} \). Simplifying further, we can rationalize to get \( 1 - \frac{\sqrt{3}}{3} \).
06

Conclusion

Thus, the value of the integral \( \int_{7\pi/6}^{5\pi/4} \sec^2 x \, dx \) is \( 1 - \frac{\sqrt{3}}{3} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Antiderivative
The concept of an antiderivative is crucial in calculus because it provides the foundation for solving integrals. In simple terms, the antiderivative of a function is another function that, when differentiated, gives the original function back. For instance, the antiderivative of the function \( \sec^2 x \) is \( \tan x \).
This is because the derivative of \( \tan x \) with respect to \( x \) is \( \sec^2 x \). It's like working backwards from what you find in derivative problems.
  • The process of finding an antiderivative is known as integration.
  • If \( F'(x) = f(x) \), then \( F(x) \) is an antiderivative of \( f(x) \).
  • Antiderivatives are essential for solving definite integrals.
Understanding this concept helps you unlock the mysteries of definite integrals and makes trigonometric integrals more approachable.
Exploring Trigonometric Integrals
Trigonometric integrals involve the integration of trigonometric functions. These types of integrals are common in calculus due to the periodic nature of trigonometric functions, which are applicable in various fields such as physics and engineering.
For our particular problem, we are working with the integral of \( \sec^2 x \). This requires knowledge of specific integral formulas, such as knowing that the integral of \( \sec^2 x \) is \( \tan x \).
  • Integral formulas for trigonometric functions can simplify complex problems.
  • Common trigonometric integrals include those involving \( \sin x \), \( \cos x \), and \( \tan x \).
  • Familiarity with these integral formulas is essential to solve or simplify trigonometric integrals effectively.
By understanding these formulas, you can confidently tackle trigonometric integrals and apply them to definite integrals.
The Art of Evaluating Limits in Definite Integrals
Evaluating limits is an integral part of solving definite integrals. The term "limits" here refers to the boundaries of integration, \( a \) and \( b \), which specify the interval over which the function is integrated.
In our exercise, we calculate the definite integral from \( 7\pi/6 \) to \( 5\pi/4 \) of \( \sec^2 x \) by substituting these limits into the antiderivative \( \tan x \).
  • First, evaluate the antiderivative at the upper limit, \( b \).
  • Next, evaluate the antiderivative at the lower limit, \( a \).
  • Finally, subtract the result of the lower limit from the upper limit to find the value of the integral.
This process, using the fundamental theorem of calculus, transforms an indefinite integral into a calculable quantity. Understanding this approach allows you to find the specific area under a curve defined by trigonometric functions within specified limits.

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Most popular questions from this chapter

For Exercises 107 and \(108,\) refer to the following: With the advent of summer come fireflies. They are intriguing because they emit a flashing luminescence that beckons their mate to them. It is known that the speed and intensity of the flashing are related to the temperature- the higher the temperature, the quicker and more intense the flashing becomes. If you ever watch a single firefly, you will see that the intensity of the flashing is periodic with time. The intensity of light emitted is measured in candelas per square meter (of firefly). To give an idea of this unit of measure, the intensity of a picture on a typical TV screen is about 450 candelas per square meter. The measurement for the intensity of the light emitted by a typical firefly at its brightest moment is about 50 candelas per square meter. Assume that a typical cycle of this flashing is 4 seconds and that the intensity is essentially zero candelas at the beginning and ending of a cycle. Graph the equation from Exercise 107 for a period of 30 seconds.

Find all the values of \(\theta, 0 \leq \theta \leq 2 \pi,\) for which the equation \(\cos \theta=\frac{1}{4} \sec \theta\) is true.

If a sound wave is represented by \(y=0.008 \sin (750 \pi t) \mathrm{cm},\) what are its amplitude and frequency? See Exercise 99.

For Exercises \(87-90\), refer to the following: Set the calculator in parametric and radian modes and let $$ \begin{array}{l} X_{1}=\cos T \\ Y_{1}=\sin T \end{array} $$ (TABLE CANNOT COPY) Set the window so that \(0 \leq \mathrm{T} \leq 2 \pi,\) step \(=\frac{\pi}{15},-2 \leq \mathrm{X} \leq 2\) and \(-2 \leq Y \leq 2 .\) To approximate the sine or cosine of a T value, use the \([\text { TRACE }]\) key, type in the T value, and read the corresponding coordinates from the screen. Approximate \(\sin \left(\frac{\pi}{3}\right),\) take 5 steps of \(\frac{\pi}{15}\) each, and read the \(y\) -coordinate.

In Exercises \(67-94,\) add the ordinates of the individual functions to graph each summed function on the indicated interval. $$y=2 \sin \left(\frac{x}{2}\right)-\cos (2 x), 0 \leq x \leq 4 \pi$$
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