Beginning CalculusApplications of Definite Integrals - Areas and Volumes -

Shahrizal Shamsuddin Norashiqin Mohd Idrus

Department of Mathematics,FSMT - UPSI

(LECTURE SLIDES SERIES)

VillaRINO DoMath, FSMT-UPSI

Applications of Definite Integrals - Areas and Volumes 1 / 9

Areas Between Curves Volumes - Method of Disks Method of Shells

Learning Outcomes

Compute the areas between to curves.

Use disk or shell methods to compute volumes.

VillaRINO DoMath, FSMT-UPSI

Applications of Definite Integrals - Areas and Volumes 2 / 9

Areas Between Curves Volumes - Method of Disks Method of Shells

Area Between Curves

y

x

dx

a b

f(x)

g(x)

A =∫ b

a[f (x)− g (x)] dx

VillaRINO DoMath, FSMT-UPSI

Applications of Definite Integrals - Areas and Volumes 3 / 9

Areas Between Curves Volumes - Method of Disks Method of Shells

Example - Method 1

Find the area between x = y2 and y = x − 2

­1 1 2 3 4 5

­4

­2

2

4

x

y

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Applications of Definite Integrals - Areas and Volumes 4 / 9

Areas Between Curves Volumes - Method of Disks Method of Shells

Volumes By Slicing

A

dx

∆V = A∆xdV = Adx

V =∫Adx

VillaRINO DoMath, FSMT-UPSI

Applications of Definite Integrals - Areas and Volumes 5 / 9

Areas Between Curves Volumes - Method of Disks Method of Shells

Solids of Revolution - Around the x-axis

y = f (x)

y

xa b

dx

y

y

xa b

y

dx

A

V =∫ b

a

(πy2

)dx

VillaRINO DoMath, FSMT-UPSI

Applications of Definite Integrals - Areas and Volumes 6 / 9

Areas Between Curves Volumes - Method of Disks Method of Shells

Example

Volume of a ball of radius a

y

xa

dx

dV = πy2dx

(x − a)2 + y2 = a2 ⇒ y2 = 2ax − x2

V =∫ 2a

0π(2ax − x2

)dx =

43

πa3unit3

VillaRINO DoMath, FSMT-UPSI

Applications of Definite Integrals - Areas and Volumes 7 / 9

Areas Between Curves Volumes - Method of Disks Method of Shells

Example - continue

V (x) := volume of portion of width x of ball.

x

V(x)

V (x) = π

(ax2 − x

3

3

)(Check) . If x = a, then

V (x) = π

(a3 − a

3

3

)=23

πa3unit3

VillaRINO DoMath, FSMT-UPSI

Applications of Definite Integrals - Areas and Volumes 8 / 9

Areas Between Curves Volumes - Method of Disks Method of Shells

Solid of Revolution - Around the y-axis

y

x

y = x2

y = a dx

y

x

y

x

y

x

dx

y

x

y

x

dx

Thickness := dxHeight := ytop − ybottom = a− y = a− x2Circumference := 2πx

dV = (2πx)(a− x2

)dx = 2π

(ax − x3

)dx

V =∫ √a0

2π(ax − x3

)dx =

12

πa2unit3

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Applications of Definite Integrals - Areas and Volumes 9 / 9