Aim: Area Between Two Curves Course: Calculus

Do Now:

Aim: How do we find the area of a region between two curves?

3

220

1Evaluate:

1 4dx

x

Aim: Area Between Two Curves Course: Calculus

Area of Region Between 2 Curves

3

2

1

1 2 3

ba

( )b

af x dx

3

2

1

1 2 3

ba

( )b

ag x dx

f(x) g(x)

3

2

1

1 2 3

ba

Area of region between f(x) and g(x)

( ) ( )b

af x g x dx

Aim: Area Between Two Curves Course: Calculus

Representative Rectangle

3

2

1

1 2 3

ba

( )ig x

( )if x

ix

f

gx

( ) ( )i if x g x height ;x width

1

Area =lim ( ) ( )n

i ini

f x g x x

( ) ( )b

af x g x dx

area of representative rectangle

height width ( ) ( )i i iA f x g x x

Aim: Area Between Two Curves Course: Calculus

Area of Region Between 2 Curves

If f and g are continuous on [a, b] and g(x) < f(x) for all x in [a, b], then the area of the region bounded by the graphs of f and g

and the vertical lines x = a and x = b is

( ) ( )b

aA f x g x dx

y

( , ( ))x g x

( , ( ))x f x

( ) ( )f x g x

True regardless of relative position of x-axis; as long as f and g are continuous.

f

g

xa b

Aim: Area Between Two Curves Course: Calculus

Model Problem

Find the area of the region bounded by the graphs of y = x2 + 2, y = -x, x = 0, and x = 1.

( ) ( )b

aA f x g x dx

1 2

02x x dx

1 2

02x x dx

13 2

0

23 2

1 1 172

3 2 6

x xx

Check with Calculator

4

3

2

1

-1

-2 2g x = -x

f x = x2+2

Aim: Area Between Two Curves Course: Calculus

Model Problem

Find the area of the region bounded by the graph of f(x) = 2 – x2 and g(x) = x.

2

1

-1

-2

-2 2

g x = xf x = 2-x2 ( ) ( )

b

aA f x g x dx

a and b ?points of intersection

1 12 2

2 22 2A x x dx x x dx

22 x x 2 or 1x

2; 1a b

13 2

2

92

3 2 2

x xx

Aim: Area Between Two Curves Course: Calculus

Model Problem

The sine and cosine curves intersect infinitely many times, bounding regions of equal areas. Find the area of one of these regions.

1.5

1

0.5

-0.5

-1

-1.5

-2 2 4

g x = cos x

f x = sin x points of intersection a and b?

sin cosx x

sin cos1

cos cos

x x

x x tan x

5 or

4 4x

5 4

4sin cosA x x dx

5 4

4cos sinx x

2 2

Aim: Area Between Two Curves Course: Calculus

Do Now:

Aim: How do we find the area of a region between two curves?

Find the area of the region between the graphs of f(x) = 1 – x2 and g(x) = 1 – x.

1

0.8

0.6

0.4

0.2

-0.2

0.5 1

g x = 1-x

f x = 1-x2

Aim: Area Between Two Curves Course: Calculus

Model Problems

Find the area of the region between the graphs of f(x) = 3x3 – x2 – 10x and

g(x) = -x2 + 2x.

5

-5

-2 2

g x = -x2+2x

f x = 3x3-x2-10x3 2 23 10 2x x x x x

33 12 0x x

23 4 0x x

0, 2x

( ) ( )f x g x ( ) ( )g x f x

0 2

2 0( ) ( ) ( ) ( )A f x g x dx g x f x dx

points of intersection

Aim: Area Between Two Curves Course: Calculus

Model Problems

Find the area of the region between the graphs of f(x) = 3x3 – x2 – 10x and

g(x) = -x2 + 2x.0 23 3

2 03 12 3 12A x x dx x x dx

0 24 42 2

2 0

3 36 6

4 4

x xx x

12 24 12 24 24

Aim: Area Between Two Curves Course: Calculus

Horizontal Representative Rectangles (Slices)

vertical rectangle

( )x f y

horizontal rectangle

( )x f y

Problem

No Problemintegrate with

respect to y

If a region is bounded by f(y) on the right and g(y) on the left at all points of the interval [c, d], then the area of the region is given by 2

1

( ) ( )y

yA f y g y dy

Aim: Area Between Two Curves Course: Calculus

Horizontal Representative Rectangles

If the graph of a function of y is a boundary of a region, it is often convenient to use representative rectangles that are horizontal and find the area by integrating with respect to y.

2

1

in variable

top curve bottom c

Vertical rectangles

urvex

x

x

A dx

2

1

in variable

right curve left cu

Horizontal rectangles

rvey

y

y

A dy

Aim: Area Between Two Curves Course: Calculus

Model Problem

Find the area of the region between the curve x = y2 and the curve x = y + 6 from y = 0 to y = 3.

4

2

-2

5 10

h y = y+6

g y = y2

right boundary: x = y + 6

left boundary: x = y2

for entire region:

2

1

( ) ( )y

yA f y g y dy

3 2

06A y y dy

32 3

0

62 3

y yy

27

2

Aim: Area Between Two Curves Course: Calculus

Representative Rectangle

2

1

in variable

right curve

Hori

left curv

zontal rectangle

e

s

y

y

y

A dy

Find the area of the region bounded by the graphs of x = 3 – y2 and x = y + 1.

2

1

-1

-2

2 4

g y = y+1

f y = 3-y2 points of intersection

Δy

f(y) is to the right of

g(y)

area of representative rectangle

2( ) ( ) 3 1A f y g x y y y y

(-1, -2)

(2, 1)

Aim: Area Between Two Curves Course: Calculus

Representative Rectangle

Find the area of the region bounded by the graphs of x = 3 – y2 and x = y + 1.

2

1

-1

-2

2 4

g y = y+1

f y = 3-y2 points of intersection

Δy

f(y) is to the right of

g(y)

2

1

( ) ( )y

yA f y g y dy

1 2

23 1A y y dy

(-1, -2)

(2, 1)

1 2

22y y dy

13 2

2

92

3 2 2

y yA y

Aim: Area Between Two Curves Course: Calculus

2

1

-1

-2

2 4

Representative Rectangle

Find the area of the region bounded by the graphs of x = 3 – y2 and x = y + 1.

Δx

2 3

1 21 3 3 3A x x dx x x dx

(-1, -2)

(2, 1)

(3, 0)

points of intersectiony = x – 1

3y x

3y x

Solve for y

y = x – 1

3y x

x-intercept – (3, 0)

Aim: Area Between Two Curves Course: Calculus

Model Problem

Find the area of the region between the curve y = sin x and the curve y = cos x from 0 to /2.

1

0.5

1 2

g x = cos x f x = sin x

2

4

Aim: Area Between Two Curves Course: Calculus

Model Problem

Find the area of the region between the curve3 and the line 0.x y y x

1

0.5

-0.5

-1

f y = y3-y

Aim: Area Between Two Curves Course: Calculus

Model Problem

Find the area of the region between the curve

and the x-axis from x = -3 to x = 3.

3 and the curve 3y x y x

2

1

-2 2

g x = 3-x 0.5 f x = x+3 0.5

Aim: Area Between Two Curves Course: Calculus

Model Problem