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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Use Exercise 22 to find the centroid of the triangle with vertices (0, 0), (a, 0), and (a, b), where a > 0 and b > 0.

To determine

To find: The centroid of triangle.

Explanation

Given data:

A vertices of triangle are (0,0) , (a,0) , (a,b) .

Formula used:

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

x¯=12ACx2dy (1)

y¯=12ACy2dx (2)

Here,

A is the area.

Write the expression for area of triangle (A) .

A=12ab

Here,

a and b are sides of triangle.

The curve C is divided into three sub-curves C1 , C2 and C3 .

Write the parametric equations of curve C1 , 0ta .

x=t (3)

y=0 (4)

Differentiate equation (3) with respect to t.

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

Differentiate equation (4) with respect to t.

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

Write the parametric equation of curve C2 , 0tb .

x=a (5)

y=t (6)

Differentiate equation (5) with respect to t.

ddt(x)=ddt(a)dxdt=0 {ddt(k)=0}dx=0dt

Differentiate equation (6) with respect to t.

ddt(y)=ddt(t)dydt=1 {ddt(t)=1}dy=dt

Write the parametric equations of curve C3 , t=a to t=0 .

x=t (7)

y=bat (8)

Differentiate equation (7) with respect to t.

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

Differentiate equation (8) with respect to t.

ddt(y)=ddt(bat)dydt=ba(1) {ddt(t)=1}dy=badt

Find the value of Cx2dy

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Chapter 16 Solutions

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