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Math 103 – Rimmer

5.6 Area between curves5.6 Area Between Curves

( ) ( )

Consider the region that lies between two curves

and and between the vertical lines

and .

S

y f x y g x

x a x b

= =

= =

( ) ( ) [ ]

Here, and are continuous functions

and for all in , .

f g

f x g x x a b≥

Math 103 – Rimmer

5.6 Area between curves

( ) ( )

We divide into strips of equal width and approximate the

th strip by a rectangle with base and height * *i i

S n

i x f x g x∆ −

2

Math 103 – Rimmer

5.6 Area between curves

[ ]1

lim ( *) ( *)n

i in

i

A f x g x x→∞

=

= − ∆∑

[ ]1

The Riemann sum

is therefore an approximation to what

we intuitively think of as the area of

(

.

*) ( *)n

i i

i

f x g x x

S

=

− ∆∑

This approximation appears to become better and better as .n → ∞

Thus, we define the area of the region as the limiting value

of the sum of the areas of these approximating rectangles.

A S

Math 103 – Rimmer

5.6 Area between curves

( ) ( )b

aA f x g x dx= − ∫

Thus, we have the following formula for area :

( ) ( )

( ) ( ) [ ]

Remember is described as the region bounded by the curves

and and the lines and ,

where, and are continuous and for all in , .

S

y f x y g x x a x b

f g f x g x x a b

= = = =

≥

the area of A S=

upper lower

function function

b

aA dx

= −

∫

3

Math 103 – Rimmer

5.6 Area between curves

[ ]( ) ( )d

cA f y g y dy= −∫

Some regions are best treated by

regarding as a function of .x y

( ) ( )

( ) ( ) [ ]

If a region is bouned by the curves and

and the lines and , where and are continuous

and for all in , , then its area is:

x f y x g y

y c y d f g

f y g y y c d

= =

= =

≥

right left

function function

d

cA dy

= −

∫

Math 103 – Rimmer

5.6 Area between curves

2

Find where the curves intersect:

2 4x x x− = +2 3 4 0x x− − =

( )( )4 1 0x x− + =

4, 1x x⇒ = = −

( ) ( )4

2

1

4 2A x x x dx−

⇒ = + − − = ∫4

2

1

3 4x x dx−

− + + = ∫4

3 2

1

34

3 2

x xx

−

−+ +

( ) ( )( )

3 23 2 1 3 14 3 44 4 4 1

3 2 3 2

− − ⋅ − − ⋅= + + ⋅ − + + ⋅ −

64 1 3

24 16 43 3 2

− = + + − + −

65 344

3 2

−= + −

130 264 9

6

− + −=

125

6=

4

Math 103 – Rimmer

5.6 Area between curves

cosy x=

sin 2y x=

Find where the curves intersect:

cos sin 2x x=

Use the trig. ident. :

sin 2 2sin cosx x x=

2sin cos cos 0x x x− =

sin 2 cos 0x x− =

( )cos 2sin 1 0x x − =

cos 0 or 2sin 1 0x x= − =

1 or sin

2 2 6x x x

π π⇒ = = ⇒ =

[ ] [ ]6 2

06

cos sin 2 sin 2 cosA x x dx x x dx

π π

π

⇒ = − + −∫ ∫

[ ] [ ]1 16 22 20

6

sin cos 2 cos 2 sinx x x xπ π

π= + + − −

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1 1 1 12 2 2 2

sin cos sin 0 cos 0 cos sin cos sin6 3 2 3 6

π π π π ππ = + − + + − − − − −

[ ] [ ] [ ] [ ]1 1 1 1 1 12 4 2 2 4 2

0 1= + − + + − − − − 1 14 4

= +1

2=

Math 103 – Rimmer

5.6 Area between curves

the integral should be done in terms of

since the lower limit will change when you reach 3

dy⇒

We need the equation of the tangent line

the curves need to be expressed as functions of y⇒2

y x x y= ⇒ =

3612 36 3

12 12

y yy x x x

+= − ⇒ = ⇒ = +

( )36

0

312

yA y dy

⇒ = + −

∫

362 3/2

0

23

24 3

y yy

= + −

2 3/236 2 363 36

24 3

⋅= + ⋅ −

1/236 2 3636 3

24 3

⋅= + −

336 3 4

2

= + −

136

2

=

18=

( )this is the slope of the tangent line. The tangency pt. is 6,36

2 2 evaluated at 6 12y x y x x y′ ′= ⇒ = = ⇒ =

( )using 36 12 6y mx b b= + ⇒ = + 36 72 36b b⇒ = + ⇒ = −

the equation of the line is 12 36y x=⇒ −

5

Math 103 – Rimmer

5.6 Area between curves