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Chapter 8: Problem 2

Evaluate the integrals in Exercises \(1-24\) using integration by parts. $$ \int \theta \cos \pi \theta d \theta $$

Short Answer

Expert verified
\( \frac{\theta}{\pi} \sin(\pi \theta) + \frac{1}{\pi^2} \cos(\pi \theta) + C \)

Step by step solution

01

Identify Parts of the Formula

Integration by parts is given by the formula \( \int u \, dv = uv - \int v \, du \). We need to choose which parts of our integral to identify as \( u \) and \( dv \). For the integral \( \int \theta \cos(\pi \theta) \, d\theta \), set \( u = \theta \) and \( dv = \cos(\pi \theta) \, d\theta \).
02

Differentiate and Integrate

Now, differentiate \( u \) and integrate \( dv \). Thus, \( du = d\theta \) and to find \( v \), integrate \( dv \): \( dv = \cos(\pi \theta) \, d\theta \), which integrates to \( v = \frac{1}{\pi} \sin(\pi \theta) \).
03

Apply Integration by Parts Formula

Apply the integration by parts formula: \( \int u \, dv = uv - \int v \, du \). Substitute \( u = \theta \), \( du = d\theta \), \( v = \frac{1}{\pi} \sin(\pi \theta) \) into the formula: \( \int \theta \cos(\pi \theta) \, d \theta = \left( \theta \cdot \frac{1}{\pi} \sin(\pi \theta) \right) - \int \left( \frac{1}{\pi} \sin(\pi \theta) \right) \, d\theta \).
04

Solve Remaining Integral

Simplify and solve the remaining integral: \( \int \sin(\pi \theta) \frac{1}{\pi} \, d\theta \). This is \( \frac{1}{\pi} \int \sin(\pi \theta) \, d\theta \), which integrates to \( -\frac{1}{\pi^2} \cos(\pi \theta) + C \), where \( C \) is the constant of integration.
05

Complete the Solution

Substitute back into the integration by parts result: \( \int \theta \cos(\pi \theta) \, d \theta = \frac{\theta}{\pi} \sin(\pi \theta) + \frac{1}{\pi^2} \cos(\pi \theta) + C \). The final answer is the integration result with the respective constants.

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Key Concepts

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

Definite Integrals
When we talk about definite integrals, we are discussing the evaluation of an integral over a specified interval on the x-axis. Unlike indefinite integrals, which include a constant of integration representing a family of functions, definite integrals yield a numerical value. This process involves determining the area under a curve from one point to another. Typically, in calculus, we specify limits for these integrals.

Definite integrals are useful in numerous applications, such as computing areas, volumes, and other accumulative quantities in physics and engineering.
  • Specified limits: A definite integral is bounded by upper and lower limits, usually denoted as \(a\) and \(b\).
  • Computation: The value of a definite integral can be found using the Fundamental Theorem of Calculus, which relates antiderivatives to definite integrals.
  • Area calculation: The result represents the net area between the curve and the x-axis, taking below-axis areas to be negative.
Understanding the concept of definite integrals is key to solving more advanced problems, where specific values rather than variable expressions are required.
Trigonometric Integration
Trigonometric integration involves integrating functions that contain trigonometric functions such as sine, cosine, tangent, etc. This type of integration is crucial when dealing with periodic phenomena like oscillations and waves.

In the integration by parts exercise we worked on, trigonometric functions played a vital role. For example, we managed the integral of \(\cos(\pi \theta)\) by applying the integration by parts technique.
  • Common techniques include using trigonometric identities to simplify or transform the integrands.
  • Sometimes, substitution is necessary in simplifying the process or converting to a more manageable integral.
  • Recognizing patterns such as \(\sin^2 x + \cos^2 x = 1\) helps in rewriting integrals into solvable forms.
Mastering trigonometric integration opens doors to solving a variety of calculus problems where wave-like behavior is modeled.
Calculus Problem Solving
Calculus problem solving extends beyond rote memorization of formulas. It demands a solid understanding of concepts and the logical application of techniques. Integration by parts is one of these methods, especially useful when integrating the product of two functions, as we have seen in the problem with \(\theta \cos(\pi \theta)\).

The process involves choosing appropriate parts of the function to differentiate and integrate, respectively. Here are some tips:
  • Select \(u\) and \(dv\) wisely to simplify the integration process.
  • Remember the formula: \(\int u \, dv = uv - \int v \, du\).
  • Verify by differentiating your final answer to see that it matches the original integrand.
Approaching calculus problems as puzzles can enhance problem-solving skills and boost confidence in tackling new and complex problems.

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Most popular questions from this chapter

Life expectancy At birth, a French citizen has an average life expectancy of 81 years with a standard deviation of 7 years. If 100 newly born French babies are selected at random, how many would you expert to live between 75 and 85 years? Assume life expectancy is normally distributed.

Find the area of the region that lies between the curves \(y=\sec x\) and \(y=\tan x\) from \(x=0\) to \(x=\pi / 2\) .

Elliptic integrals The length of the ellipse $$ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 $$ turns out to be $$Length =4 a \int_{0}^{\pi / 2} \sqrt{1-e^{2} \cos ^{2} t} d t$$ where \(e=\sqrt{a^{2}-b^{2}} / a\) is the ellipse's eccentricity. The integral in this formula, called an elliptic integral, is nonelementary except when \(e=0\) or \(1 .\) a. Use the Trapezoidal Rule with \(n=10\) to estimate the length of the ellipse when \(a=1\) and \(e=1 / 2\) . b. Use the fact that the absolute value of the second derivative of \(f(t)=\sqrt{1-e^{2} \cos ^{2} t}\) is less than 1 to find an upper bound for the error in the estimate you obtained in part (a).

Airport waiting time According to the U.S. Customs and Border Protection Agency, the average airport wait time at Chicago's O'Hare International airport is 16 minutes for a traveler arriving during the hours \(7-8\) A.M., and 32 minutes for arrival during the hours \(4-5\) P.M. The wait time is defined as the total processing time from arrival at the airport until the completion of a passen- ger's security screening. Assume the wait time is exponentially distributed. $$ \begin{array}{l}{\text { a. What is the probability of waiting between } 10 \text { and } 30 \text { minutes }} \\ {\text { for a traveler arriving during the } 7-8 \text { A.M. hour? }}\\\\{\text { b. What is the probability of waiting more than } 25 \text { minutes for a }} \\ {\text { traveler arriving during the } 7-8 \text { A.M. hour? }} \\ {\text { c. What is the probability of waiting between } 35 \text { and } 50 \text { minutes }} \\\ {\text { for a traveler arriving during the } 4-5 \text { P.M. hour? }}\\\\{\text { d. What is the probability of waiting less than } 20 \text { minutes for a }} \\ {\text { traveler arriving during the } 4-5 \text { P.M. hour? }}\end{array} $$

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. $$\int_{0}^{\pi / 2} \tan \theta d \theta$$
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