The experimental verification of a theory concerning any natural phenomenon generally rests on the...

The experimental verification of a theory The experimental verification of a theory concerning any natural phenomenon concerning any natural phenomenon generally rests on the result of an generally rests on the result of an integration…integration…

J. W. Mellor…J. W. Mellor…

AP CalculusAP Calculus Area of a Region Between Two CurvesArea of a Region Between Two Curves

Larson – Hostetler – Edwards: Chapter 7.1Larson – Hostetler – Edwards: Chapter 7.1

§ Area of a Region Between two Curves Area of a Region Between two Curves

If If ff and and gg are continuous on are continuous on [[a, ba, b]] and and gg ((xx) ) f f ((xx)) for for all all xx in in [[a, ba, b],], then the area of the region bounded by the then the area of the region bounded by the graphs of graphs of ff and and gg and the vertical lines and the vertical lines x = ax = a and and x = bx = b isis

dxxgxfAb

a )()(

(Upper Function – Lower Function)(Upper Function – Lower Function)



§ Area of a Region Between two CurvesArea of a Region Between two Curves

dxxfb

a )(f f ((xx))

dxxgb

a )(

gg ((xx))



§ Area of a Region Between two CurvesArea of a Region Between two Curves

dxxfb

a )(

f f ((xx))

dxxgb

a )(

g g ((xx))

f f ((xx))

g g ((xx)) dxxgxf

b

a )()(



§ Example.Example. Find the area between the two given curvesFind the area between the two given curves

2).( xyxya

From: From: Paul's Online Math Notes

http://tutorial.math.lamar.edu/classes/calcI/areabetweencurves.aspx








2,01).(2

xxxeyxyb x










102164).( 2 xyxyc










5,2102164).( 2 xxxyxyd










2/,0sincos).( xxxyxye










132

1).( 2 yxyxf










22 )2(10).( yxyxg









PracticePractice.. Textbook 7.1 pp. 452-455, prob. 1, 3, 5, 13, 17, Textbook 7.1 pp. 452-455, prob. 1, 3, 5, 13, 17, 21, 23, 27, 33, 37, 39, 43, 45, 4721, 23, 27, 33, 37, 39, 43, 45, 47

Suggested Extra Practice Suggested Extra Practice : : pp 452-455, prob. 2, 4, 6, pp 452-455, prob. 2, 4, 6, 14, 18, 22, 24, 28, 34, 38, 40, 14, 18, 22, 24, 28, 34, 38, 40, 44, 46, 4844, 46, 48

The moving power of mathematical The moving power of mathematical invention is not reasoning but invention is not reasoning but imaginationimagination ……

Augustus de Morgan…Augustus de Morgan…

This is a solid obtained by rotating a region in the plane This is a solid obtained by rotating a region in the plane about an axis (called axis of revolution). about an axis (called axis of revolution).

§ Solids of RevolutionSolids of Revolution

AP CalculusAP Calculus Volume: The Disk MethodVolume: The Disk Method




This is a solid obtained by rotating a region in the plane This is a solid obtained by rotating a region in the plane about an axis (called axis of revolution). about an axis (called axis of revolution).





The simplest solid of revolution is a The simplest solid of revolution is a right circular cylinder or disk, which is right circular cylinder or disk, which is formed by revolving a rectangle about formed by revolving a rectangle about an adjacent axisan adjacent axis

ww

rr

ww

rr

The corresponding volume is:The corresponding volume is:

Volume disk = area disk Volume disk = area disk width disk width disk

V = V = rr22 ww

r = r = radius of the disk radius of the disk w =w = widthwidth




To find the volume of any To find the volume of any solid of revolution, consider solid of revolution, consider this as composed of this as composed of nn-disks-disks of width of width ww = = xx and radius and radius rr = = r r (( xxi i ))

An approximation for the An approximation for the volume of the solid will be:volume of the solid will be:

xxrVn

ii

2

1

])([




Increasing the numberIncreasing the number nn of of disks ( i. e. disks ( i. e. xx 0 0 ), the ), the approximation to the volume approximation to the volume becomes better. becomes better.

The volume of the solid will The volume of the solid will be given by the limit process:be given by the limit process:

xxrVn

ii

2

10

])([lim

b

adxxr 2)]([

§ The Disk MethodThe Disk Method



If If rr((xx)) is continuous and is continuous and r(xr(x) > 0) > 0 on [a, b], then the on [a, b], then the volume obtained by rotating the region under the graph, volume obtained by rotating the region under the graph, can be calculated using one of the following formulas. can be calculated using one of the following formulas.

Horizontal Axis of Revolution:Horizontal Axis of Revolution: b

adxxrV 2)]([

Vertical Axis of Revolution:Vertical Axis of Revolution: b

adyyrV 2)]([



§ Example.Example. Show that the volume of a sphere of radius r isShow that the volume of a sphere of radius r is

22 xry xx

yy

00 rr rr

b

adxxyV 2)]([

xx

yy

3

3

4rV



§ Example.Example. Show that the volume of a right cone of radius Show that the volume of a right cone of radius r r and height and height hh is is given by the formula :given by the formula :

xh

ry

b

adxxyV 2)]([

xx

yy

00 hh

rr

hrV 2

3

1

xx

yy



§ Example.Example. Find the volume of the solid obtained by rotating about the Find the volume of the solid obtained by rotating about the xx-axis -axis the region between the region between x = x = 00, x = , x = 11, and under the curve , and under the curve

xy



§ Example.Example. Find the volume of the solid obtained by rotating about the Find the volume of the solid obtained by rotating about the xx-axis -axis the region between the lines the region between the lines x = x = 1,1, x = x = 3,3, and under the curve and under the curve

xy

1



§ Example.Example. Find the volume of the solid obtained by rotating the region Find the volume of the solid obtained by rotating the region bounded by bounded by y = xy = x 33, y = , y = 88,, and and x = x = 0 0 about the about the yy-axis-axis

3xy

b

adyyxV 2)]([

33

1

yyx



§ Example.Example. Find the volume of the solid obtained by rotating about the Find the volume of the solid obtained by rotating about the yy-axis -axis the region bounded bythe region bounded by x = x = 44,, x = x = 0 0 andand

xy

b

adyyxV 2)]([

2yx



PracticePractice.. Textbook 7.2 pp. 463-466, prob. 1, 3, 5, 7, Textbook 7.2 pp. 463-466, prob. 1, 3, 5, 7, 9, 11, 139, 11, 13

Suggested Extra Practice Suggested Extra Practice : : pp 463-466, prob. 2, 4, pp 463-466, prob. 2, 4, 6, 8, 10, 12, 146, 8, 10, 12, 14

§ The Washer MethodThe Washer Method



Method useful to find the volume of a solid of revolution Method useful to find the volume of a solid of revolution with a with a holehole, by changing the , by changing the diskdisk with a with a washerwasher. The . The washer is formed by revolving a rectangle between to washer is formed by revolving a rectangle between to curves around the curves around the x-axisx-axis or any axis of the form or any axis of the form x = ax = a

ww

RRrr

ww

RRrr

V V = = ((RR22 – – rr22))ww

Volume Volume of of

WasherWasher

§ The Washer MethodThe Washer Method



IfIf R R((xx)) and r and r((xx)) are continuous functions on are continuous functions on [[a,a, bb]], , RR((xx) > 0) > 0, , rr((xx) > 0,) > 0, and and RR((xx) > ) > rr((xx),), then the volume then the volume obtained by rotating the region between obtained by rotating the region between RR((xx)) and and rr((xx)) can be calculated using one of the following formulas:can be calculated using one of the following formulas:

Horizontal Axis Horizontal Axis of Revolution:of Revolution:

b

adxxrxRV 22 )]([)]([

Vertical Axis of Vertical Axis of Revolution:Revolution: dyyryRV

b

a 22 )]([)]([



§ Example.Example. Find the volume of the solid obtained by rotating the region Find the volume of the solid obtained by rotating the region between the lines between the lines ff((xx)) = = xx22 + + 33,, and and gg((xx)) = x = x22 + + 11, around the , around the x- x- axisaxis

3)( 2 xxR

b


1)( 2 xxr



§ Example.Example. Find the volume of the solid obtained by rotating the region Find the volume of the solid obtained by rotating the region between the lines between the lines ff((xx)) and and gg((xx)), around the , around the x- x- axisaxis

212)( xxf

b


2)( xg



§ Example.Example. Find the volume of the solid obtained by rotating the region Find the volume of the solid obtained by rotating the region between the lines between the lines ff((xx)) and and gg((xx)), around the axis , around the axis y = y = 11

2)( 2 xxf

b


24)( xxg



§ Example.Example. Find the volume of the solid obtained by rotating the region Find the volume of the solid obtained by rotating the region between the lines between the lines ff((xx),), gg((xx) and ) and x x = 1= 1, around the axis , around the axis y = y = 22

4)( xxf

b


( ) / 3g x x



§ Example.Example. Find the volume of the solid formed by revolving the region Find the volume of the solid formed by revolving the region bounded by the graph of bounded by the graph of y = xy = x22 + + 11, y = , y = 00 , x = , x = 00 and and x = x = 11, , about the about the yy - axis - axis

b

adyyryRV 22 )]([)]([



§ Example.Example. Find the volume of the solid formed by revolving the region Find the volume of the solid formed by revolving the region bounded by the graph of bounded by the graph of y = y = 3/3/x, x, andand y = y = 4 4 x, x, about about x = x = 11

b

adyyryRV 22 )]([)]([



PracticePractice.. Textbook 7.2 pp. 463-466, prob. 15, 17, 19, 21, Textbook 7.2 pp. 463-466, prob. 15, 17, 19, 21, 23, 27, 2923, 27, 29

Suggested Extra Practice Suggested Extra Practice : : pp 463-466, prob. 16, pp 463-466, prob. 16, 18, 20, 22, 24, 28, 3018, 20, 22, 24, 28, 30

§ Solids with Known Cross SectionsSolids with Known Cross Sections



With the disk method it is possible to find the volume of a With the disk method it is possible to find the volume of a solid with a circular cross section: solid with a circular cross section:

RR((xx)])]2 2 == AA((xx)) b

adxxRV 2)]([

b

adxxA )(

This method can be generalized to solids with any shape, This method can be generalized to solids with any shape, as long as a formula for the area of the arbitrary cross as long as a formula for the area of the arbitrary cross section is knownsection is known

§ Solids with Known Cross SectionsSolids with Known Cross Sections



Area of Cross SectionArea of Cross Section == AA((xx))

xxAVn

ii

1

)(

b

adxxAV )(

n

ii

xxxAV

10

)(lim

§ Volumes of Solids with Known Cross SectionsVolumes of Solids with Known Cross Sections



1. For cross sections of area 1. For cross sections of area AA((xx)) taken perpendicular to taken perpendicular to the the xx-axis: -axis:

b

adxxAV )(

2. For cross section of area 2. For cross section of area AA((yy)) taken perpendicular taken perpendicular to the to the yy-axis-axis

b

adyyAV )(



§ Example.Example. Find the volume of the solid with semicircular cross section and Find the volume of the solid with semicircular cross section and whose base is bounded by the graphs of whose base is bounded by the graphs of y = y = 00, x = , x = 99 andand

b

adxxAV )(

xy

From: From: DEMOS with POSITIVE IMPACT, NSF DUE 9952306 DEMOS with POSITIVE IMPACT, NSF DUE 9952306

xy

rr

2

xr

2

22

1)(

xxA



§ Example.Example. Find the volume of the solid with square cross section and whose Find the volume of the solid with square cross section and whose base is bounded by the graphs of base is bounded by the graphs of

b

adxxAV )(

1,3/,3/ xxyxy

2

3

2)(

x

xA

2

3

x

2

3

x

From: From: DEMOS with POSITIVE IMPACT DEMOS with POSITIVE IMPACT

David R. Hill, Temple University - Lila F. Roberts, Georgia College & State UniversityDavid R. Hill, Temple University - Lila F. Roberts, Georgia College & State University



§ Example.Example. Find the volume of the solid with equilateral triangle cross section Find the volume of the solid with equilateral triangle cross section and base bounded by the graphs of and base bounded by the graphs of y = sin x, x = y = sin x, x = 00, x = , x =

b

adxxAV )(

xxxA sin

2

3sin

2

1)(

xsin

sin x





§ Example.Example. Find the volume of the solid with square cross section and base Find the volume of the solid with square cross section and base bounded by the graphs of bounded by the graphs of x = x = 00, x = , x = yy = 0 = 0 andand

b

adxxAV )(

221)( xxA

21 x

21 x

21 xy





§ Example.Example. Find the volume of the solid with equilateral triangle cross section Find the volume of the solid with equilateral triangle cross section and base bounded by the circle of radius and base bounded by the circle of radius 22, centered at , centered at (0, 0)(0, 0)

b

adxxAV )( 243)( xxA



xx

yy 24 xy

24 xy 242 x

243 x



PracticePractice.. Textbook 7.2 pp. 463-466, prob. 61, 62, 63, 64 Textbook 7.2 pp. 463-466, prob. 61, 62, 63, 64

Suggested Extra Practice Suggested Extra Practice : : pp 463-466, prob. 61, pp 463-466, prob. 61, 62, 63, 6462, 63, 64

The advancement and perfection of The advancement and perfection of mathematics are intimately connected mathematics are intimately connected with the prosperity of the State …with the prosperity of the State …

Napoleon…Napoleon…

This is an alternative method for finding the volume of a This is an alternative method for finding the volume of a solid of revolutionsolid of revolution

Compared with the disk method, the representative Compared with the disk method, the representative rectangular section is parallel to the axis of revolutionrectangular section is parallel to the axis of revolution

§ The Shell MethodThe Shell Method

AP CalculusAP Calculus Volume: The Shell MethodVolume: The Shell Method

Disk MethodDisk Method


ww

rr

ww

rr

Shell MethodShell Method

hh

wwpp

hh

pp




Disk MethodDisk MethodShell MethodShell Method





Volume Volume of Shell:of Shell:

hw

phw

p22

22

whp2

hh

wwpp

yy

xx

pp

hh

p + wp + w // 22

p p w w // 22ww

yy

xx

If If ww = = yy,, p p = = pp((yy), ), h h = = hh((yy)) V V = 2= 2[[ pp((yy)) hh((yy)])]yy




The volume of the solid obtained by revolving the area The volume of the solid obtained by revolving the area under function under function hh, can be calculated in the next form:, can be calculated in the next form:

Vertical Vertical axis of axis of

revolutionrevolution

d

cdyyhypV )()(2Horizontal Horizontal

axis of axis of revolutionrevolution

b

adxxhxpV )()(2

pp((yy)) = = distance to axisdistance to axis

pp((xx)) = = distance to axisdistance to axis



§ Example.Example. Find the volume of the solid obtained by rotating the area between Find the volume of the solid obtained by rotating the area between the graph of the graph of ff((xx)) = = 11 – – 22x – x – 33xx22 – – 22xx33 and the and the xx-axis over -axis over [0, 1] [0, 1] about: about: ((aa)) the the yy-axis, -axis, ((bb) ) xx = = 11

Rogawski, 402Rogawski, 402

((aa)) ((bb))

b

adxxhxpV )()(2



§ Example.Example. Calculate the volume of the solid obtained by rotating the area Calculate the volume of the solid obtained by rotating the area enclosed by the graphs of enclosed by the graphs of ff((xx)) = = 99 – x – x22 and and gg((xx)) = = 99 – – 33xx about: about: ((aa)) x-x-axis, (b) y = 10axis, (b) y = 10

Rogawski, 403Rogawski, 403

((aa)) ((bb))



§ Example.Example. Calculate the volume of the solid obtained by rotating the region Calculate the volume of the solid obtained by rotating the region between the graph between the graph ff ((yy)) = = 12(12(yy22 – y – y33)) and theand the y- y-axis,axis, over over [0, 1], [0, 1], about: about: ((aa)) the the xx-axis, -axis, ((bb)) the line the line y y = = 11

Finney, 392Finney, 392

((aa)) ((bb))



§ Example.Example. Calculate the volume of the solid obtained by rotating the region Calculate the volume of the solid obtained by rotating the region between the graphs of between the graphs of f f ((xx)) = y = y 4

4/4 – /4 – yy 22/2 /2 and and x = yx = y 2

2//22 about: about: (a) (a) x-x-axis, (b) axis, (b) y = y = 22

Stewart 395Stewart 395

((aa)) ((bb))



§ Example.Example. Calculate the volume of the solid obtained by rotating the region Calculate the volume of the solid obtained by rotating the region between the graph between the graph f f ((xx)) = x sin = x sin x x and theand the x- x-axisaxis,, over over [0, [0, ]] about: about: (a)(a) the the y-y-axis, axis, x = x = 22

Stewart 396Stewart 396

((aa)) ((bb))



§ Example.Example. Calculate the volume of the solid obtained by rotating the region Calculate the volume of the solid obtained by rotating the region between the graph between the graph f f ((xx)) = sin = sin x x and:and:

((aa)) thethe x- x-axisaxis,, over over [0, [0, ]] about the about the line x = line x = 55((bb) ) the line the line y = y = 44, over [0, 4], about the , over [0, 4], about the yy-axis-axis

((bb))((aa))



§ Example.Example. Calculate the volume of the solid obtained by rotating the circle Calculate the volume of the solid obtained by rotating the circle ((x – x – 2)2)22 + y + y22 = = 11 about the about the y-y-axisaxis



PracticePractice.. Textbook 7.2 pp. 463-466, prob. 1, 3, 5, 7, 9, 11, Textbook 7.2 pp. 463-466, prob. 1, 3, 5, 7, 9, 11, 14, 15, 17, 1914, 15, 17, 19

Suggested Extra Practice Suggested Extra Practice : : pp 463-466, prob. 2, 4, 6, pp 463-466, prob. 2, 4, 6, 8, 10, 12, 14, 16, 18, 208, 10, 12, 14, 16, 18, 20







Horizontal Horizontal axis of axis of


b

adxxhxpV )()(2


d

cdyyhypV )()(2








d

cdyyhypV )()(2

Horizontal Horizontal axis of axis of



b

adxxhxpV )()(2


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