Math 5A: Area Under a Curve. The Problem: Find the area of the region below the curve f(x) = x 2 +1...

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

TIFF (Uncompressed) decompressorare needed to see this picture.

IDEAS?

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

IDEA

Approximate Area by:

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

= 6

Actual Area _______ 6



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.



4 rectangles

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

functional value. Area of rectangles=

Actual Area ____3.75



€

=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



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



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



More Rectangles-Left Endpoint







# 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




are needed to see this picture.QuickTime™ and aTIFF (Uncompressed) decompressor


Estimate Appears to Approach 4.6666…Regardless of Point Chosen

Left Endpoint Right Endpoint Midpoint







Choose Random Point



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

∑



Remember the big step?

Date post:	16-Jan-2016
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Math 5A: Area Under a Curve. The Problem: Find the area of the region below the curve f(x) = x 2 +1...

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