/*! 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 16 Sketch the region bounded by the... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 16

Sketch the region bounded by the graphs of the algebraic functions and find the area of the region. $$ f(x)=-x^{2}+4 x+1, g(x)=x+1 $$

Short Answer

Expert verified
The area of the region bounded by the two given functions is 9 square units.

Step by step solution

01

Finding Intersection Points

First, set the two functions equal to each other and solve for x. So, solve \(f(x) = g(x)\), or \(-x^2 + 4x + 1 = x + 1\). This simplifies to \(x^2 - 3x = 0\), which factoring x out gives \(x(x - 3) = 0\). From this, it's deduced that the two functions intersect at \(x = 0\) and \(x = 3\).
02

Drawing the Graph

Next, the region of interest is between \(x=0\) and \(x=3\). By plotting both functions, it can be seen that \(f(x)\) is above \(g(x)\) between these two points.
03

Calculating the Area

The area of the region between these two functions is given by the definite integral \(\int_{a}^{b} (f(x) - g(x)) dx\), where [a,b] is the interval on which the point of intersection lie. Here, calculate \(\int_{0}^{3} ((-x^2 + 4x + 1) - (x + 1)) dx\), which simplifies to \(\int_{0}^{3} (-x^2 + 3x) dx\). The integral of \(-x^2\) with respect to x is \(-\frac{1}{3}x^3\) and the integral of \(3x\) is \(\frac{3}{2}x^2\). Now, substitute the upper and lower limits of integration: \(-\frac{1}{3}(3)^3 + \frac{3}{2}(3)^2 - (-\frac{1}{3}(0)^3 + \frac{3}{2}(0)^2)\). This simplifies to \(9\).
04

Final Answer

So, the area of the region bounded by the two functions is 9 square units.

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

Use the disk method to verify that the volume of a sphere is \(\frac{4}{3} \pi r^{3}\).

(a) use a graphing utility to graph the region bounded by the graphs of the equations, \((b)\) find the area of the region, and (c) use the integration capabilities of the graphing utility to verify your results. $$ f(x)=6 x /\left(x^{2}+1\right), \quad y=0, \quad 0 \leq x \leq 3 $$

Sketch the region bounded by the graphs of the algebraic functions and find the area of the region. $$ f(y)=\frac{y}{\sqrt{16-y^{2}}}, \quad g(y)=0, \quad y=3 $$

Fluid Force on a Circular Plate A circular plate of radius \(r\) feet is submerged vertically in a tank of fluid that weighs \(w\) pounds per cubic foot. The center of the circle is \(k(k>r)\) feet below the surface of the fluid. Show that the fluid force on the surface of the plate is \(F=w k\left(\pi r^{2}\right)\) (Evaluate one integral by a geometric formula and the other by observing that the integrand is an odd function.)

In Exercises \(27-34,\) (a) use a graphing utility to graph the region bounded by the graphs of the equations, \((b)\) find the area of the region, and (c) use the integration capabilities of the graphing utility to verify your results. $$ f(x)=x\left(x^{2}-3 x+3\right), \quad g(x)=x^{2} $$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics 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.