Math 5A: Area Under a Curve

The Problem: Find the area of the

region below the curve f(x) = x2+1 over the interval [0, 2]. QuickTime™ and a

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IDEAS?

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IDEA

Approximate Area by:

Area Triangle + Area Rectangle = (2)(1) + (1/2)(2)(4)

= 6

Actual Area _______ 6

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IDEA: Vertical strips Cut the region into

vertical strips. Cut the top

horizontally to make rectangles.

Approximate the area under the curve using the rectangles.

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4 rectangles

Each rectangle Width= = =1/2 Height determined by

functional value. Area of rectangles=

Actual Area ____3.75

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=1 12( ) + 5

412( ) + 2 1

2( ) + 134

12( ) = 3.75

€

f (0)Δx + f 12( )Δx + f 1( )Δx + f 3

2( )Δx

€

B − A

4

€

Δx

4 rectangles-right endpoint

Each rectangle Width= = =1/2 Height determined by

functional value at rt endpt. Area of rectangles=

Actual Area ____5.75€

f 12( )Δx + f 1( )Δx + f 3

2( )Δx + f (2)Δx€

Δx

€

B − A

4

€

= 54

12( ) + 2 1

2( ) + 134

12( ) + 5 1

2( ) = 5.75

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Caution Left endpoint does not always yield an

underestimate, nor right an overestimate. Consider f(x) = sinx on [0, ]

4 rectangles-midpoint

Each rectangle Width= = =1/2 Height determined by functional

value at midpoint. Area of rectangles=

Actual Area ____4.625€

f 14( )Δx + f 3

4( )Δx + f 54( )Δx + f 7

4( )Δx€

Δx

€

B − A

4

€

=1716

12( ) + 25

1612( ) + 41

1612( ) + 65

1612( ) = 4.625

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Improving Area Estimate

Left Endpoint - 8 Rectangles Width = = =1/4 Area of Rectangles

€

Δx

€

B − A

8

€

f 0( )Δx + f 14( )Δx + f 1

2( )Δx + f ( 34 )Δx +

f 1( )Δx + f 54( )Δx + f 3

2( )Δx + f ( 74 )Δx =

4.1875

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More Rectangles-Left Endpoint

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# rect=16 32 64

Area= 4.421875 4.542969 4.604492

More Rectangles-Right Endpoint

# rect= 16 32 64

Area=4.921875 4.792969 4.729492

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Estimate Appears to Approach 4.6666…Regardless of Point Chosen

Left Endpoint Right Endpoint Midpoint

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Choose Random Point

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Best Estimate As the number of rectangles, n,

increases, the area in the rectangles appears to approach the area under the curve, regardless of point chosen.

DefineArea Under the Curve =

€

limn→∞

(AreaRectangles)

Generalizing… f(x) conts. and f(x) 0 on [a,b] Partition [a,b]

a random point in

€

≥

€

a = x0 < x1 < ...< xn = b

€

x i*

€

[x i−1,x i ]

€

Δx =b− a

n

€

AREA = limn→∞

f (x i*)Δx

i=1

n

∑

Formalize Example

If choose right endpoint

€

Δx =b− a

n=

€

i 2n( )

€

=limn→∞

i 2n( )( )

2+1[ ]

i=1

n

∑ 2

n€

f (x i*) = f i 2

n( )( ) = i 2n( )( )

2+1

€

AREA = limn→∞

f (x i*)Δx

i=1

n

∑

€

2

n

€

x i* = x i = a+ iΔx =

Example - continued

€

L =limn→∞

i 2n( )( )

2+1[ ]

i=1

n

∑ 2

n=

€

=limn→∞

8

n3i2 +

2

ni=1

n

∑i=1

n

∑ ⎡

⎣ ⎢

⎤

⎦ ⎥=

€

limn→∞

8

n3i2 +

2

n

⎡ ⎣ ⎢

⎤ ⎦ ⎥

i=1

n

∑

€

limn→∞

8

n3

n(n +1)(2n +1)

6

⎛

⎝ ⎜

⎞

⎠ ⎟ +

2

nn

⎡

⎣ ⎢

⎤

⎦ ⎥

BIG STEP HERE !!What happened?

Example continued

€

L =limn→∞

8

n3

n(n +1)(2n +1)

6

⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢

⎤

⎦ ⎥+ limn→∞

2

nn

⎡ ⎣ ⎢

⎤ ⎦ ⎥

So…AREA =

€

4 23

€

=16

6+ 2 =

14

3

Corresponds to earlier approximation of 4.6666….

Things to Think About What if f(x) <0 on all or part of [a,b]?

Can we always find a closed form expression for ?

What do we do when we can’t?

€

f (x i*)Δx

i=1

n

∑

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Remember the big step?