/*! 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 20 Use any method to evaluate the i... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 20

Use any method to evaluate the integrals in Exercises \(15-34 .\) Most will require trigonometric substitutions, but some can be evaluated by other methods. $$ \int \frac{\sqrt{9-w^{2}}}{w^{2}} d w $$

Short Answer

Expert verified
The integral is \(-\frac{\sqrt{9-w^2}}{w} - \arcsin \left(\frac{w}{3}\right) + C\).

Step by step solution

01

Identify the Integral Form

The given integral is \( \int \frac{\sqrt{9-w^{2}}}{w^{2}} \, dw \). This is a type of integral where trigonometric substitution can be utilized, specifically involving square roots of the form \( \sqrt{a^2 - b^2} \). The presence of a square root in the numerator and a polynomial in the denominator suggests using a trigonometric substitution.
02

Use Trigonometric Substitution

Substitute \( w = 3\sin\theta \), which implies \( dw = 3\cos\theta \, d\theta \). With this substitution, the square root becomes \( \sqrt{9 - w^2} = \sqrt{9 - 9\sin^2\theta} = 3\cos\theta \). The integral transforms as follows:\[ \int \frac{\sqrt{9-w^{2}}}{w^{2}} \, dw = \int \frac{3\cos\theta}{(3\sin\theta)^2} \cdot 3\cos\theta \, d\theta \] which simplifies to:\[ \int \frac{3\cos\theta}{9\sin^2\theta} \cdot 3\cos\theta \, d\theta = \int \frac{9\cos^2\theta}{9\sin^2\theta} \, d\theta \] This further simplifies to:\[ \int \cot^2 \theta \, d\theta \]
03

Simplify the Integral

The integral \( \int \cot^2 \theta \, d\theta \) can be simplified using the identity \( \cot^2 \theta = \csc^2 \theta - 1 \), which gives:\[ \int \cot^2 \theta \, d\theta = \int (\csc^2 \theta - 1) \, d\theta \] This integral can now be separated and evaluated:\[ \int \csc^2 \theta \, d\theta - \int 1 \, d\theta = - \cot \theta - \theta + C \] where \( C \) is the constant of integration.
04

Back-Substitute the Trigonometric Functions

Recall that \( w = 3\sin\theta \), thus \( \sin\theta = \frac{w}{3} \) and \( \cos\theta = \sqrt{1 - \sin^2\theta} = \sqrt{1 - \left(\frac{w}{3}\right)^2} = \frac{\sqrt{9-w^2}}{3} \). Thus, \( \cot\theta = \frac{\cos\theta}{\sin\theta} = \frac{\sqrt{9-w^2}}{w} \).Substituting back in terms of \( w \), we have:\[ - \cot \theta - \theta = - \frac{\sqrt{9-w^2}}{w} - \arcsin \left(\frac{w}{3}\right) \] Thus, the integral in terms of \( w \) becomes:\[ - \frac{\sqrt{9-w^2}}{w} - \arcsin \left(\frac{w}{3}\right) + C \]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Integration Techniques
Integration techniques are various methods used to find the integral of a function. One powerful method is trigonometric substitution. This can greatly simplify integrals involving square roots or quadratic expressions. Here, we identify the integral, recognizing the structure with a square root like \( \sqrt{9 - w^2} \), and apply a substitution.
  • Substitution involves setting \( w = 3\sin\theta \) because it matches the format \( a^2 - b^2 \).
  • By substituting, the square root simplifies due to the identity \( \sin^2 \theta + \cos^2 \theta = 1 \).
This transformation turns a complicated integral into a more manageable form, enabling us to use basic integration techniques in an altered trigonometric space.
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables where they are defined. They are essential tools in integration and calculus. During trigonometric substitution, identities help simplify expressions to integrate. For example, the identity \( \cot^2 \theta = \csc^2 \theta - 1 \) is useful here.
  • This identity breaks down \( \cot^2 \theta \) into simpler terms that are easily integrated.
  • The identity \( \csc^2 \theta = 1 + \cot^2 \theta \) also plays a critical role in simplifying expressions.
Using these identities allows us to convert complex trigonometric forms into integrals that can be solved directly.
Calculus Problem Solving
Solving calculus problems often involves recognizing patterns and substituting appropriately to simplify the problem. In the given integral, we begin by identifying the problem's structure and deciding on trigonometric substitution as the solution.
  • Determine the substitution needed by analyzing the square root structure in the original integral.
  • Simplify the expression using known trigonometric identities and regularly review results to ensure all simplifications are valid.
Successful problem solving in calculus combines understanding advanced techniques and using them to break down complex problems into simpler tasks.
Definite and Indefinite Integrals
Integrals are a fundamental concept in calculus and can be classified as definite or indefinite. An indefinite integral includes a constant of integration \( C \), representing an entire family of solutions. This exercise explores the indefinite integral of the given function.
  • Indefinite integrals express general solutions without specific boundaries. It鈥檚 crucial to include the constant \( C \) in the final answer.
  • Definite integrals, on the other hand, provide a specific numerical value by using specific limits of integration.
An understanding of both types is essential, as solutions often depend on whether a general form or a specific value is required.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Lifetime of a tire Assume the random variable \(L\) in Example 2\(f\) is normally distributed with mean \(\mu=22,000\) miles and \(\sigma=4,000\) miles. a. In a batch of 4000 tires, how many can be expected to last for at least \(18,000\) miles? b. What is the minimum number of miles you would expect to find as the lifetime for 90\(\%\) of the tires?

\begin{equation} \begin{array}{l}{\text { Exercises } 71-74 \text { are about the infinite region in the first quadrant }} \\ {\text { between the curve } y=e^{-x} \text { and the } x \text { -axis. }}\end{array} \end{equation} Find the centroid of the region.

Printer lifetime The lifetime of a \(\$ 200\) printer is exponentially distributed with a mean of 2 years. The manufacturer agrees to pay a full refund to a buyer if the printer fails during the first year following its purchase, and a one-half refund if it fails during the second year. If the manufacturer sells 100 printers, how much should it expect to pay in refunds?

In Exercises 65 and \(66,\) use a CAS to perform the integrations. Evaluate the integrals $$ (a)\int x \ln x d x \quad \text { b. } \int x^{2} \ln x d x \quad \text { c. } \int x^{3} \ln x d x $$ $$ \begin{array}{l}{\text { d. What pattern do you see? Predict the formula for } \int x^{4} \ln x d x} \\ {\text { and then see if you are correct by evaluating it with a CAS. }} \\ {\text { e. What is the formula for } \int x^{n} \ln x d x, n \geq 1 ? \text { Check your }} \\ {\text { answer using a CAS. }}\end{array} $$

Suppose you toss a fair coin \(n\) times and record the number of heads that land. Assume that \(n\) is large and approximate the discrete random variable \(X\) with a continuous random variable that is normally distributed with \(\mu=n / 2\) and \(\sigma=\sqrt{n} / 2 .\) If \(n=400,\) find the given probabilities. a. \(P(190 \leq X<210) \quad\) b. \(P(X<170)\) c. \(P(X>220) \quad\) d. \(P(X=300)\)
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.