/*! 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 7 The integrand of the definite in... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 7

The integrand of the definite integral is a difference of two functions. Sketch the graph of each function and shade the region whose area is represented by the integral. $$ \int_{0}^{4}\left[(x+1)-\frac{1}{2} x\right] d x $$

Short Answer

Expert verified
The area under the curve represented by the integral \(\int_{0}^{4}(x+1-\frac{1}{2}x) dx\) from 0 to 4 is 8 units square.

Step by step solution

01

Identify the Functions

The two functions in the integrand are \(f(x) = x+1\) and \(g(x) = -\frac{1}{2}x\). These will be graphed separately to later find the area between them from 0 to 4.
02

Sketch the Graphs of the Functions

Graph the first function \(f(x) = x + 1\) which is a line with a slope of 1 and crosses the y-axis at 1. Then, graph the second function \(g(x) = -\frac{1}{2}x\), a line with negative slope -1/2 and crosses the y-axis at 0. Both lines cross at the point (2,2).
03

Shade the Region

The area represented by the integral is the region between these two graphs from x=0 to x=4. Shade this region to show the area.
04

Evaluate the Definite Integral

The definite integral \(\int_{0}^{4}(x+1-\frac{1}{2}x) dx\) can now be solved. Simplify the integrand to \(\int_{0}^{4}(0.5x+1)dx\), separate and integrate term-by-term to obtain \(0.5*\frac{1}{2}*x^2+x |_{0}^{4}\) = 0.5*8+4-0 = 8.
05

Interpretation

The value obtained from the integral, which is 8, corresponds to the area of the shaded region on the graph. This area is enclosed by the two curves from x=0 to x=4 and is equal to the given integral value.

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 Midpoint Rule with \(n=4\) to approximate the area of the region bounded by the graph of \(f\) and the \(x\) -axis over the interval. Compare your result with the exact area. Sketch the region. $$ f(x)=4-x^{2} \quad[-2,2] $$

Use a graphing utility to graph the region bounded by the graphs of the functions, and find the area of the region. $$ f(x)=-x^{2}+4 x+2, g(x)=x+2 $$

The velocity \(v\) of the flow of blood at a distance \(r\) from the center of an artery of radius \(R\) can be modeled by \(v=k\left(R^{2}-r^{2}\right), \quad k>0\) where \(k\) is a constant. Find the average velocity along a radius of the artery. (Use 0 and \(R\) as the limits of integration.)

Sketch the region bounded by the graphs of the functions and find the area of the region. $$ y=\frac{1}{x}, y=x^{3}, x=\frac{1}{2}, x=1 $$

The integrand of the definite integral is a difference of two functions. Sketch the graph of each function and shade the region whose area is represented by the integral. $$ \int_{-1}^{1}\left[\left(1-x^{2}\right)-\left(x^{2}-1\right)\right] d x $$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Statistics

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.