/*! 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 22 Find the area of the region boun... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 22

Find the area of the region bounded by the curves. \(y=x^{2}-1\) and \(y=3\)

Short Answer

Expert verified
The area is 16.

Step by step solution

01

Identify the points of intersection

Set the equations of the curves equal to each other to find the points of intersection: y = x^2 - 1 = 3.Solve for x:x^2 - 1 = 3,x^2 = 4,x = 卤2.So, the points of intersection are (-2, 3) and (2, 3).
02

Set up the integral

The area between the curves can be found by integrating the difference of the functions from -2 to 2: \[\int_{-2}^{2} (3 - (x^2 - 1)) \, dx\].
03

Simplify the integrand

Simplify the integrand to make the integral easier: 3 - (x^2 - 1) = 4 - x^2. Thus, the integral becomes: \[\int_{-2}^{2} (4 - x^2) \, dx\].
04

Integrate the function

Integrate \(4 - x^2\) with respect to x: \[\int_{-2}^{2} (4 - x^2) \, dx = \left[ 4x - \frac{x^3}{3} \right]_{-2}^{2}\].
05

Evaluate the definite integral

Plug in the upper and lower bounds into the antiderivative and subtract: \[\left( 4(2) - \frac{2^3}{3} \right) - \left( 4(-2) - \frac{(-2)^3}{3} \right)\].Simplify:\[\left( 8 - \frac{8}{3} \right) - \left( -8 + \frac{-8}{3} \right)\],= \( \left( \frac{24}{3} - \frac{8}{3} \right) - \left( -\frac{24}{3} - \frac{8}{3} \right) \),= \( \frac{16}{3} + \frac{32}{3} \),= \( \frac{48}{3} \),=16.

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.

definite integral
A definite integral is used to find the area under a curve between two specific points on the x-axis. When we calculate a definite integral, we are essentially summing up an infinite number of infinitesimally small areas to get the total area. In our exercise, we need the area between the curves \( y=x^2-1 \) and \( y=3 \) from \( x = -2 \) to \( x = 2 \). This involves integrating the difference between the two functions within the given bounds. The integral we need to solve is: \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ oI Our simple integrand in this example makes finding the area straightforward.
points of intersection
Identifying points of intersection between the given curves is crucial. These points indicate where the curves meet, and they establish the limits for our definite integral. With our equations, \( y = x^2 - 1 \) and \( y = 3 \), we set them equal to find where they intersect: \ \ \ \

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

The velocity at time \(t\) seconds of a ball thrown up into the air is \(v(t)=-32 t+75\) feet per second. (a) Find the displacement of the ball during the time interval \(0 \leq t \leq 3\). (b) Given that the initial position of the ball is \(s(0)=6\) feet, use (a) to determine its position at time \(t=3 .\)

Find the real number \(b>0\) so that the area under the graph of \(y=x^{2}\) from 0 to \(b\) is equal to the area under the graph of \(y=x^{3}\) from 0 to \(b\).

Determine \(\Delta x\) and the midpoints of the subintervals formed by partitioning the given interval into \(n\) subintervals. $$1 \leq x \leq 4 ; n=5$$

Determine the average value of \(f(x)\) over the interval from \(x=a\) to \(x=b\), where $$f(x)=1 / x ; a=1 / 3, b=3$$

Use a graphing utility to find the intersection points of the curves, and then use the utility to find the area of the region bounded by the curves. $$y=1 / x, y=3-x$$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

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.