   Chapter 16.4, Problem 23E

Chapter
Section
Textbook Problem

Use Exercise 22 to find the centroid of a quarter-circular region of radius a.

To determine

To find: The centroid of the quarter-circular region of radius a.

Explanation

Given data:

Formula used:

Write the expression for centroid (x¯,y¯) .

x¯=12ACx2dy (1)

y¯=12ACy2dx (2)

The region of quarter-circle region is shown in Figure 1.

Consider a quarter-circle as C. From Figure 1, the curve C is,

C=C1+C2+C3

Here,

C1,C2andC3 are curves.

Consider the curves as,

C1:x=t,y=0,0ta;C2:x=acost,y=asint,0tπ2;C3:x=0,y=at,0ta;

Find the value of dx for curve C1 .

ddt(x)=ddt(t)dxdt=1 {ddx(x)=1}dx=dt

Find the value of dy for curve C1 .

ddt(y)=ddt(0)dydt=0 {ddx(0)=0}dy=0dt

Find the value of dx for curve C2 .

ddt(x)=ddt(acost)dxdt=a(sint) {ddx(cosx)=sinx}dx=asintdt

Find the value of dy for curve C2 .

ddt(y)=ddt(asint)dydt=acost {ddx(sint)=cost}dy=acostdt

Find the value of dx for curve C3 .

ddt(x)=ddt(0)dxdt=0 {ddx(0)=0}dx=0dt

Find the value of dy for curve C3 .

ddt(y)=ddt(at)dydt=01 {ddx(k)=0,ddx(x)=1}dy=dt

Write the expression for area of quarter-circle (A) .

A=14πr2

Here,

Substitute a for r,

A=14πa2

Substitute 14πa2 for A and C1+C2+C3 for C in equation (1),

x¯=12(14πa2)C1+C2+C3x2dy=1(12πa2)C1+C2+C3x2dy

x¯=2πa2(C1x2dy+C2x2dy+C3x2dy) (3)

Find the value of C1x2dy+C2x2dy+C3x2dy .

C1x2dy+C2x2dy+C3x2dy=0a(t)2(0dt)+0π2(acost)2(acostdt)+0a(0)2(dt)=0+0π2(a3cos3t)dt+0=a30π2(cos2t)costdt=a30π2(1sin2t)costdt{cos2t=1sin2t}

C1x2dy+C2x2dy+C3x2dy=a30π2(costsin2tcost)dt=a3[sint13sin3t]0π2=a3[(sin(π2)13sin3(π2))(sin(0)13sin3(0))]=a3[(113(1)3)(013(0)3)]

Simplify the equation

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Convert from degrees to radians. 1. 210

Single Variable Calculus: Early Transcendentals, Volume I

Solve the equations in Exercises 112 for x (mentally, if possible). 7x+55=98

Finite Mathematics and Applied Calculus (MindTap Course List)

A sample with a mean of M = 8 has X = 56. How many scores are in the sample?

Essentials of Statistics for The Behavioral Sciences (MindTap Course List)

In problem 18 and 19, determine whether each graph represents as a function of . 19.

Mathematical Applications for the Management, Life, and Social Sciences

Given I=EZ,E=240, and Z=15, find I.

Elementary Technical Mathematics

limnn2+3n2n2+n+1= a) 0 b) 12 c) 1 d)

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

True or False: Du f(a, b) = ∇(a, b) · u.

Study Guide for Stewart's Multivariable Calculus, 8th 