Log In Start studying!

Select your language

Suggested languages for you:
Deutsch (DE)
Deutsch (UK)
Deutsch (US)

Americas

Europe

Answers without the blur. Sign up and see all textbooks for free! → Illustration

Q18E

Expert-verified
Essential Calculus: Early Transcendentals
Found in: Page 369
Essential Calculus: Early Transcendentals

Essential Calculus: Early Transcendentals

Book edition 2nd
Author(s) James Stewart
Pages 830 pages
ISBN 9781133112280

Answers without the blur.

Just sign up for free and you're in.

Register for Free I’ll do it later
Illustration

Short Answer

Sketch the region enclosed by the given curves and

find its area. 18. \(y = \left| x \right|,y = {x^2} - 2\).

The \(Area = \frac{{20}}{3}\).

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

The sketch of the graph of given equations

As always, draw a picture or use calculator.

A tip for using TI-83 or TI-84 graphing calculator, you can graph the absolute value function by pressing the “Y=” button, then the “MATH” button, then scroll right to the “NUM” category. Select “abs(“ and enter the variable x to graph the function appropriately.

Find the intersection points

To find where the intersection points are, we must set the equations equal to one another and solve for x. However, because we are interacting with the absolute value function, we must be careful while moving ahead.

Whenever dealing with the absolute value function in these problems, always be aware that there are two cases.

\(\left| x \right| = {x^2} - 2\)

When x is greater than zero

\(\begin{aligned}{l}x = {x^2} - 2\\0 = {x^2} - x - 2\\x = - 1\;or\;x = 2\end{aligned}\)

When x is less than zero

\(\begin{aligned}{l} - x = {x^2} - 2\\0 = {x^2} + x - 2\\x = 1\;or\;x = - 2\end{aligned}\)

Check derived answer against the graph by finding area

\(\begin{aligned}{l}A = 2\int\limits_0^2 {(x - ({x^2} - 2))dx} \\ = 2\left( {\left( {\frac{{ - {{\left( 2 \right)}^3}}}{3} + \frac{{{{\left( 2 \right)}^2}}}{2} + 2(2)} \right) - (0)} \right)\\ = 2\left( {\frac{{ - 8}}{3} + 2 + 4} \right)\\ = 2\left( {\frac{{10}}{3}} \right)\\ = \frac{{20}}{3}\end{aligned}\)

Most popular questions for Math Textbooks

Graph the region between the curves and use your calculator to compute the area correct to five decimal places.

\(y = {\tan ^2}x,\quad y = \sqrt x \)

(a) Find the number \(a\) such that the line \(x = a\) bisects the area under the curve y = 1/ x2, 1 ≤ x ≤ 4

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer

\({y^2} = x,\,x = 2y\)about \(y\)-axis

To determinethe volume generated by rotating the region bounded by the given curve by the use of the method of cylindrical shell.

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

\(y = 1 - {x^2},y = 0;\) about the \(x\)-axis.

Icon

Want to see more solutions like these?

Sign up for free to discover our expert answers
Get Started - It’s free

Recommended explanations on Math Textbooks

Decision Maths

Read explanation

Calculus

Read explanation

Probability and Statistics

Read explanation

Statistics

Read explanation

Mechanics Maths

Read explanation

Geometry

Read explanation

Pure Maths

Read explanation

94% of StudySmarter users get better grades.

Sign up for free
94% of StudySmarter users get better grades.