math 27 lecture guide chapter 3

8/4/2019 Math 27 Lecture Guide Chapter 3

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UNIT 3. APPLICATIONS OF THE DEFINITE INTEGRAL MATH 27 LECTURE GUIDE

Objectives: By the end of the unit, a student should be able to apply methods of integration in finding: volume of the solids of revolution centroids of plane area and solids of revolution length of arc of a curve, and

area of surfaces of revolution.

MUST !!! It is expected that you already know how to evaluate definite integrals. Most examples willonly require setting up the definite integral that will solve for the measure of the object required. Incase you forgot, this is how definite integrals are evaluated:

__________________________

3.1 Area of a Plane Region (TC7 pp. 389-396 / TCWAG pp. 352-359)

Using vertical strips, Using horizontal strips,

d

c

a b

Area: Area:

If f is continuous on b,a and CxFdxxf , then aFbFdxxfb

a .

REMEMBER!!! To get area of a plane region, integrate area of a strip. Use the length of a the

strip for the length and dx or dy for the width.

Length Width

Vertical strip: xgxf ( belowabove ) dx

Horizontal strip: ygyf ( leftright ) dy

TO DO!!! Determine the area of the region bounded

by xsiny and xcosy ,4

5

4

x .

xfy

xgy

yfx

ygx


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3.2 Volume of a Solid of Revolution Using Disks or Washers(TC7 pp. 398-414 / TCWAG pp. 374-387)

Volume of a cylinder = (area of a cross section) x (height)

Consider the solid S on the right.

Let b,axi and ixA be the area of cross-section of the object at ix .

Let xi be a certain thickness of the cross-

section at ix . (This cross-section is perpendicular

to the x axis.)

Volume of a slice =

Appoximate volume of the solid S =(using several slices)

Volume of solid S = Volume of solid S =(as a Riemann sum) (as a definite integral)

where xA is an area of a cross-section.

If the cross-sections that will be used are perpendicular to the y axis, the volume of a solid will be

given by

d

cdyyAV , where yA is an area of a cross-section and dy is the thickness of a slice.

EXERCISE. Determine the area of the following region. Try using vertical and horizontal strips.1. the region bounded by xsiny and xcosy ,

4

7

4

x

2. the region bounded by xy and 3xy

3. the region bounded by xey , xlny , ex1

TO DO!!! Determine the area of the region

bounded by xey and xy 2 , 31 y .


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To get the volume of a solid of revolution using DISKS ,

consider a strip of the region that is perpendicular withthe axis of revolution. (Revolving this strip generates adisk.)

If the strips are vertical,

b

a

dxxAV where xA is an area of a cross-section

Hence, b

a

dxxrV 2

where xr is the radius of a cross-section.

If the strips are horizontal, d

c

dyyrV 2 where yr is the radius of a cross-section.

TO DO!!! Consider the solid formed by revolving

the region bounded by xy , the x axis and the

line 4x about the x axis .

A solid of revolution is generated by revolving a plane region about an axis of revolutionthat is

either tangent to the region or does not pass through the region. The resulting solid has a circularcross-section: an area of 2r with the center on the axis of revolution.

TO DO!!! Let R be the region bounded byxcosy , the x axis and the y axis.

Using disks, set-up the definite integral that willsolve for the volume of the solid generated byrevolving R about

a. the x axis

b. the y axis

xfy


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To get the volume of a solid of revolution (with a cavity) using WASHERS ,consider a strip of theregion that is perpendicular with the axis of revolution. (Revolving this strip generates a disk.)

TO DO!!! Let R be the region bounded by 3xy ,

and the lines 1x and 1y .

Using disks, set-up the definite integral that willsolve for the volume of the solid generated by

revolving R about

a. 1y

V

b. 1x

V


b

aio dxxrxrV

22 where xro

is the outer radius of a cross-section and

xri is the inner radius of a cross-section.

If the strips are horizontal,

d

cio dyyryrV

22 where yro

is the outer radius of a cross-section and

yri is the inner radius of a cross-section.

TO DO!!! Let R be the region bounded by

42 xy and 42 xy .

Using washers, set-up the definite integral that willsolve for the volume of the solid generated byrevolving R about

a. the y axis

V


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3.3 Volume of a Solid of Revolution Using Cylindrical Shells(TC7 pp. 398-414 / TCWAG pp. 374-387)

Volume of a cylindrical shell = hrr 2

where2

io rrr

or average radius, and

r is the thickness of the shell

Consider a strip of the region that is parallel with the axis of revolution. (Revolving this strip generatesa shell.)

EXERCISE. Set-up the definite integral that will solve for the volume of the solid of revolutiongenerated by revolving the region with the respective axis of revolution. Use disks or washers.

1. the region bounded by xy , the line 2y and the y axis revolved about

a. the y axis b. 2y c. 1x

2. the region bounded by xsiny

and xcosy

, 4

7

4

x revolved about

a. 1y b. 2y

3. the region bounded by xey , xlny , ex1 revolved about

a. the x axis b. the y axis


b

a

dxxhxrV 2 where xr is the

distance of a strip from the axis of revolution,and xh is the height of a strip.

If the strips are horizontal,

d

c

dyyhyrV 2 where yr is the

distance of a strip fromt the axis of revolution,and yh is the length of a strip.

b. the x axis V

c. the line 3x V

d. the line 2y V


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3.4 Centroid of a Plane Region (TC7 pp. 541-556 / TCWAG pp. 394-406)

Center of mass along a line (discrete case)

Consider several weights distributed along a line.

0 1 2 3 4 5 6 7 8 9 10

EXERCISE. In the exercise items given in page 5, instead of using disks or washers, usecylindrical shells in setting-up the definite integral that will solve for the volume of the solids ofrevolution.

TO DO!!! Let R be the region bounded by 42 xy and 42 xy . Using cylindrical shells,

set-up the definite integral that will solve for the volume of the solid generated by revolving R about

a. the y axis

V

b. the x axis

V

c. 2x

V

d. 4y

V

e. 2x

V


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Let m be the total mass and M be the moment of mass which is a product of a mass and its

directed distance from a point. If x is the coordinate of the center of mass along the line, then

n

i

i

n

i

ii

m

xm

m

Mx

1

1

Center of mass of a rod

In a homogeneous rod, the mass is directly proportional to the length: Lkm where k is theconstant linear density and L is the length of the rod.

In a non-homogeneous rod, the mass is directly proportional to the length: Lxm where x is the linear density at point x on the rod.

If x is the linear density of a rod, then the total mass is L

dxxm

0

and the moment of

mass is L

dxxxM

0

.

Center of mass of a plane region

Consider a region on the xy plane.

Assumptions: the region is a lamina (the object is thin sheet) and of constant area density, In effect, the center of mass will just be a function of the area of the region and its distances from thex axis and the y axis. (No mass in here!) In this case, the center of mass is a centroid of the

plane region.

Consider the region bounded by xfy and xgy , bxa . Suppose the area density is .(Simplification: constant area density)

TO DO!!!The length of a rod is 9 meters and the linear density of the rod at a point x meters from one end

is 14 x kg/m. Determine the center of mass of the rod.


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Total mass = AM Moments of mass:

axisxthefromcetandisAMx

axisythefromcetandisAMy

Center of mass:M

Mx

y

M

My x

Using vertical strips,

b

a

dxxgxfM b

ay dxxgxfxM

b

ax dxxgxf

xgxfM

2

In general, using horizontal or vertical strip,

b

astripaofthicknessLM stripoflength:L

dyordx:stripaofthickness

b

aaxisyy stripofthicknessLdaveM

b

a

axisxx stripaofthicknessLdaveM

xfy

xgy

TO DO!!! Determine the centroid of the following regions.

1. the region bounded by 2xy and 3xy

2. the region bounded by xsiny and xcosy ,4

5

4

x


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3.5 Centroid of a Solid of Revolution

Point in the three-dimensional space:

z,y,x :x directed distance from the yz plane:y directed distance from the xz plane

:z directed distance from the xy plane

Consider a region R on the xy plane revolved

about a certain axis of revolution on the xy plane.

Let be the (constant) density of the solid.

Centroid of the solid: z,y,x

M

Mx

yz

M

My xz

M

Mz

xy

VolumeM

planeproperthefromcetandisdirectedVolumemassofMoment

Remark: The centroid of a solid of revolution always lies on the axis of revolution.

Hence, 0z and either of x or y is dependent on the axis of revolution.

Caution! Be careful in the choice of the strips: either horizontal or vertical.

3. the region bounded by 2 xy , 2xy and the x axis

EXERCISE. Determine the centroid of the following regions.1. the region bounded by xy 2 and 2 xy (intersections: 11, and 24 , )

2. the region bounded by xey , xlny , ex1

y

xz


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3.6 Length of an Arc of a Curve (TC7 pp. 534-540 / TCWAG pp. 388-393)

Consider a curve defined by xfy that is

continuous and differentiable (smooth) on b,a .

The length of the arc given by xfy over b,a

is given by b

a

dxx'fL 21 .

If an arc is defined by the curve ygx for d,cy , then the length is

d

cdyy'gL

2

1 .

TO DO!!! Consider the solid generated by revolving

the region R bounded by 2xy and xy .

Determine the centroid of the solid.

1. axis of revolution: 1y ; using vertical strips

2. axis of revolution: 2x ; using horizontal strips

EXERCISE. Determine the centroid of the solid generated by revolving the same region as aboveabout the line 2x . Try using vertical and horizontal strips.

TO DO!!!1. Use definite integral to show that the circumference of a

circle 222 ryx , where r is a positive constant, is r2 .


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3.7 Area of a Surface of Revolution

Surface area of a cylinder = lr2

Consider the surface generated by revolving the

arc defined by xfy , b,ax about ahorizontal axis.

Using vertical strips, the area of the resulting surface

is given by

b

adxx'fxrSA

212 , where

xr is the radius of revolution.

If a surface is generated by revolving ygx , d,cy about a vertical axis, using horizontal strips,

the area of the resulting surface is given by d

c

dyy'gyrSA 212 , where yr is the radius

of revolution.

TO DO!!! Set-up the definite integral that will solve for the length of the arcs given by thefollowing. If the resulting integral is easy to evaluate, determine the length.

2. 2xy , 20 x

3. xcosy , x0

TO DO!!!1. Use definite integral to show that the surface area of a

sphere generated by revolving 222 ryx , where r is a

positive constant, about the x axis is 24 r .


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END OF UNIT 3 Lecture Guide

TO DO!!! Set-up the definite integral that will solve for the surface area of the surface generatedby revolving the given arc about the respective axis. If the resulting integral is easy to evaluate,determine the length.

2. the arc: xcosy , x0 ; the axis of revolution: 1y

3. the arc: 2xy , 20 x ; the axis of revolution: y axis

EXERCISE.

1. Consider the arc xey , 10 x . Determine the length of the arc. Also, determine the area

of the surface generated by revolving the arc about the y axis.2. Consider the arc xy , 31 x . Determine the area of the surface (a cone) generated by

revolving the arc about i.) the line 3x , ii) the line 2y .

math 27 lecture guide chapter 3

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