/*! 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} Q. 38 For each function f and interval... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Q. 38 (page 340)

For each function f and interval [a, b] in Exercises 34鈥�38, it is possible to find the exact signed area between the graph of f and the x-axis on [a, b] geometrically by using the areas of circles,

triangles, and rectangles. Find this exact area, and then calculate the left, right, midpoint, upper, lower, and trapezoid sums with n = 4. Which approximation rule is most accurate?

fx=3+4-x2,a,b=-2,2

Short Answer

Expert verified

The actual area of the semicircle is 21 square units

Step by step solution

01

Step 1. Given Information

The given function is fx=3+4-x2,a,b=-2,2

The function shows semicircle 3 units above from origin

02

Step 2. Finding area of semicircle

The semicircle has 16 full square and 10 partial square.

So, area is16+12脳10=21square units.

03

Step 2. Left Sum

鈭�x=b-an=2+24=1xk=a+k鈭�x=-2+kxk-1=-2+k-1=-3+k

fxk-1=3+4-k-32

The left sum=鈭�k=1nfxk-1鈭�x

=鈭�k=143+4-k-32=17.46

04

Step 4. Right Sum

The Right sum is

=鈭�k=1nfxk鈭�x

=鈭�k=143+4-k-22=17.46

05

Step 5. Midpoint sum

xk-1+xk2=k-2+k-32=2k-52

f2k-52=3+4-2k-522

The midpoint sum is

=鈭�k=14f2k-52鈭�x=鈭�k=143+4-2k-522=16.88

06

Step 6. Upper sum

The upper sum is

鈭�k=1nfMk鈭�x

f-1鈭�x+f0鈭�x+f1鈭�x+f2鈭�x=4.73+5+4.73+3=17.46

07

Step 7. Lower Sum

The lower sum is

鈭�k=1nfmk鈭�x

f-2鈭�x+f-1鈭�x+f0鈭�x+f1鈭�x=3+4.73+5+4.73=17.46

08

Step 8. Trapezoid Sum

fxk-1+fxk2=3+4-k-32+3+4-k-222

The midpoint sum is

鈭�k=143+4-k-32+3+4-k-222鈭�x=27+3

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

Determine which of the limit of sums in Exercises 47鈥�52 are infinite and which are finite. For each limit of sums that is finite, compute its value

limn鈫�鈭�鈭�k=1n(k2+k+1)

Use integration formulas to solve each integral in Exercises 21鈥�62. You may have to use algebra, educated guess and-check, and/or recognize an integrand as the result of a product, quotient, or chain rule calculation. Check each of your answers by differentiating .

鈭�23x+1dx

Calculate the exact value of each definite integral in Exercises 47鈥�52 by using properties of definite integrals and the formulas in Theorem 4.13.

鈭�13(x+1)2dx

Consider the general sigma notation 鈭�k=mnak.What do we mean when we say that ak is a function of k?

Use the Fundamental Theorem of Calculus to find the exact values of the given definite integrals. Use a graph to check your answer.

鈭�24鈥�3e2x鈭�4dx
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Calculus

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.