04/19/23 Perkins

AP Calculus AB

Day 5Section 4.2

x

1

left endptn

i

f x

a

Area Under a CurveFind the area of the region bounded by y = f(x), the x-axis, x = a, and x = b.

b

4

2

5a b

4

2

5

We call it the Lower Sum. We call it the Upper Sum.

Approximate the area by creating rectangles of equal width whose

endpoints are on f(x).

x

Each right endpoint is on f(x)Each left endpoint is on f(x)

This over-estimates the area under the curve…

This under-estimates the area under the curve…

1

n

i

f a i x x

1

right endptn

i

f x

1

1n

i

f a i x x

n = # of rectangles

1

height widthn

i

A

Each method is called a Riemann Sum.

lower sum actual area upper sum

lim lower sum actual area lim upper sumn n

1

limn

ni

f a i x x

How do we make these approximations for the area under a curve more accurate?

1

lim right endptn

ni

f x

Use more rectangles.

(Always choose whichever sum involves right endpoints.)

The Limit Definition for finding the area under a curve:

or

where b a

xn

Find the area beneath (above the x-axis) in the interval [1,3].

heightA x

2 3y x

a. Use 1 rectangle.8

6

4

2

2 4

3 2f 9 218

b. Use 2 rectangles.

8

6

4

2

2 4

3 12

1x

3 11

2x

2 3A f x f x

7 1 9 1 16

If a specific number of rectangles is given, it is often easier to find the area without using sigma!

Find the area beneath (above the x-axis) in the interval [1,3].

2 3y x

c. Use the limit definition.8

6

4

2

2 4

1

limn

ni

A f a i x x

3 1x

n

n

right endpt a i x 2

1 in

2

n

21

i

n

2right endpt 2 1 3inf

45

i

n

1

4 2lim 5

n

ni

i

n n

1 3

1

4 2lim 5

n

ni

iA

n n

1

2 4lim 5

n

ni

i

n n

1 1

2 4 2lim 5

n n

ni i

i

n n n

21 1

8 2lim 5

n n

ni i

in n

2

18 2lim 5

2n

n nn

n n

2

2

8lim 10

2n

n n

n

2

2

8 8lim lim 10

2n n

n n

n

4 10

14

Perkins

AP Calculus AB

Day 5Section 4.2

a

Area Under a CurveFind the area of the region bounded by y = f(x), the x-axis, x = a, and x = b.

b

4

2

5a b

4

2

5

Approximate the area by creating rectangles of equal width whose

endpoints are on f(x).

How do we make these approximations for the area under a curve more accurate?

The Limit Definition for finding the area under a curve:

Find the area beneath (above the x-axis) in the interval [1,3].

2 3y x

a. Use 1 rectangle.8

6

4

2

2 4

b. Use 2 rectangles.

8

6

4

2

2 4

Find the area beneath (above the x-axis) in the interval [1,3].

2 3y x

c. Use the limit definition.8

6

4

2

2 4