/*! 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 44 Area Find the area of the region... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 44

Area Find the area of the region bounded by the graphs of \(y=7 /\left(16-x^{2}\right)\) and \(y=1\)

Short Answer

Expert verified
The area of the region bounded by the curves \(y=7/(16-x^{2})\) and \(y=1\) is approximately \(2(\sqrt{7} + ln(4+\sqrt{7}))\).

Step by step solution

01

Find the x-values where the graphs intersect

To find where the graphs intersect, set the two equations equal to each other and solve for \(x\):\r\n\[y = 7/(16-x^{2}) = 1\]Solving this will result in two valid x-values as follows: solving \(7/(16-x^{2}) = 1\) we find that \(16 - x^{2} = 7\), and thus the valid x-values are \(x = -3, 3\).
02

Determine which curve is above the other

To determine which curve lies above the other in the region of interest, evaluate both functions at a number between the two found intersection points. For example, at \(x=0\), \(y=7/(16-0) = 7/16 < 1\). Thus, the graph of \(y=7/(16-x^{2})\) is below the graph of \(y=1\).
03

Calculate the area

The area of the region bounded by these curves is the integral, from -3 to 3, of the top function minus the bottom function. This translates to: 鈭玙(-3)^(3) (1 - 7/(16-x^{2})) dx. The definite integral yields \(2(\sqrt{7} + ln(4+\sqrt{7}))\).

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!

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

Continuous Function In Exercises 101 and \(102,\) find the value of \(c\) that makes the function continuous at \(x=0\) . $$ f(x)=\left\\{\begin{array}{ll}{\frac{4 x-2 \sin 2 x}{2 x^{3}},} & {x \neq 0} \\\ {c,} & {x=0}\end{array}\right. $$

Writing (a) The improper integrals $$ \int_{1}^{\infty} \frac{1}{x} d x \text { and } \int_{1}^{\infty} \frac{1}{x^{2}} d x $$ diverge and converge, respectively. Describe the essential differences between the integrands that cause one integral to converge and the other to diverge. (b) Sketch a graph of the function \(y=(\sin x) / x\) over the interval \((1, \infty) .\) Use your knowledge of the definite integral to make an inference as to whether the integral $$\int_{1}^{\infty} \frac{\sin x}{x} d x$$ converges. Give reasons for your answer. (c) Use one iteration of integration by parts on the integral in part (b) to determine its divergence or convergence.

Extended Mean Value Theorem In Exercises \(91-94\) , apply the Extended Mean Value Theorem to the functions \(f\) and \(g\) on the given interval. Find all values \(c\) in the interval \((a, b)\) such that $$\frac{f^{\prime}(c)}{g^{\prime}(c)}=\frac{f(b)-f(a)}{g(b)-g(a)}$$ $$ f(x)=\sin x, \quad g(x)=\cos x \quad\left[0, \frac{\pi}{2}\right] $$

Proof Prove the following generalization of the Mean Value Theorem. If \(f\) is twice differentiable on the closed interval \([a, b],\) then $$f(b)-f(a)=f^{\prime}(a)(b-a)-\int_{a}^{b} f^{\prime \prime}(t)(t-b) d t$$

Evaluating an Improper Integral In Exercises \(33-48\) determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. $$ \int_{3}^{6} \frac{1}{\sqrt{36-x^{2}}} d x $$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

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.