Area Approximation Methods

Using Geometer’s Sketchpad to Support Mathematical Thinking
Area Approximation Methods
How can we find an explicit value for the area
A⋅x 3 +B⋅x 2 +C⋅x+D
10
under the curve of a non-negative continuous
8
function on a specific interval? The classic
6
approach is to define a set of rectangles such
4
that the height of each rectangle is determined
2
by the value of the curve and the width of the
A
-2
B
2
rectangle is a partition of the interval. The
greater the number of rectangles, the closer the sum of their areas is to the
actual area under the curve. This is referred to as the Rectangle Approximation
Method (RAM).
To find the area under the curve of a function y = f ( x ) between x = a and
x = b , we divide the interval into n pieces. These pieces each have width
b−a
€
a + i ⋅ Δx . Now
and we call the right endpoint
€ of these subintervals
n
€
we form a rectangle whose height is f ( a + i ⋅ Δx ) and whose area is
Δx =
€
€
€
f ( a + i ⋅ Δx ) ⋅ Δx . If we sum all the areas of these rectangles,
we have
n
€A =
∑ f ( a + i ⋅ Δx) ⋅ Δx .
n
€
i=1
Finally, as we consider more and more rectangles, the quantity An gives a
better and better approximation so that we may write the area as
€
n
A = lim ∑ f ( a + i ⋅ Δx ) ⋅ Δx . €
n →∞ i=1
Geometer’s Sketchpad allows us to dynamically explore four different methods
of approximation: €
RRAM (right endpoint), LRAM (left endpoint), MRAM
(midpoint), and TAM (trapezoid).
Adapted from Exploring Calculus with The Geometer’s Sketchpad 4.0
http://www.keypress.com/catalog/products/software/Prod_GSPModExpCalc.html
Shelly Berman
p. 1 of 7
Area Approx Methods.doc
Jo Ann Fricker
Using Geometer’s Sketchpad to Support Mathematical Thinking
RRAM – Right Endpoint Rectangle Approximation Method
12
f(x ) = A⋅x 3 +B⋅x 2 +C⋅x+D
Color Controls
Set Interval
A = 0.00
B = 1.00
A
B
C = 0.00
D = 2.00
10
8
C
Area right
D
= 20.00
6
Animate n
4
n=5
x A = -3.00
x B = 2.00
h = 1.00
-8
2
-6
A
-4
Reset X-axis X-control
Reset Y-axis Y-control
B
-2
2
4
-2
Intermediate Information
-4
For the function f ( x ) = 2 + x 2 , the area under the curve using n rectangles in
the RRAMn f ( x ) can be approximated by:
An = RRAMn f ( x )
€
n
= ∑ f ( xi ) ⋅ Δx
€
i=1
n
(
= ∑ 2 + xi
i=1
n
(
2
) ⋅ Δx
= ∑ 2 + ( a + i ⋅ Δx )
i=1
n
2
) ⋅ Δx
(
2
(
2
= ∑ 2 + a2 + 2ai ⋅ Δx + i2 ⋅ ( Δx )
i=1
n
) ⋅ Δx
= ∑ 2 ⋅ Δx + a2 ⋅ Δx + 2ai ⋅ ( Δx ) + i2 ⋅ ( Δx )
i=1
3
)
Over the years, people have discovered a variety of formulas for the values of
finite sums. The most famous of these are the formula for the first n integers
€
Shelly Berman
p. 2 of 7
Area Approx Methods.doc
Jo Ann Fricker
Using Geometer’s Sketchpad to Support Mathematical Thinking
(which Gauss discovered at age 5) and the formulas for the sums of the
squares and cubes of the first n integers.
n
The first n integers:
∑i =
i=1
n
The first n squares:
€
The first n cubes:
€
n ( n + 1)
2
∑ i2 =
i=1
n
n ( n + 1) ( 2n + 1)
6
 n ( n + 1)  2
∑i =  2 


i=1
3
€
Now using the appropriate summation formula, our area for the n rectangles is
approximated by:
(
An = RRAMn 2 + x 2
)
 n ( n + 1)   b − a  2  n ( n + 1) ( 2n + 1)  b − a  3
 b − a
2  b − a
= n ⋅ 2⋅ 
⋅

+ n⋅ a ⋅
 + 2a
 +

2
6
 n 
 n 

  n 

 n 
 n + 1
 1  ( n + 1) ( 2n + 1) 
2
3
= 2( b − a) + a2 ⋅ ( b − a) + a
 ⋅ ( b − a)
 ⋅ ( b − a) +  
n⋅n
 n 
 6 

€
Applying this formula to the interval a = −3 to b = 2 for n = 5 rectangles:
(
A5 = RRAM5 2 + x 2
)


€  5 + 1 €
2€  1  ( 5 + 1) ( 2( 5) + 1)
3
2
 ⋅ ( 2 − ( −3) )
= 2( 2 − ( −3) ) + ( −3) ⋅ ( 2 − ( −3) ) + ( −3) ⋅ 
 ⋅ ( 2 − ( −3) ) +  

5⋅ 5
 5 
 6 

 6  2  1  6  11 3
= 2( 5) + 9( 5) + ( −3) ( 5) +    ( 5)
 5
 6  5  5 
= 10 + 45 − 90 + 55
= 20
Now, taking the limit of this sum as n → ∞ gives:
€
€
Shelly Berman
p. 3 of 7
Area Approx Methods.doc
Jo Ann Fricker
Using Geometer’s Sketchpad to Support Mathematical Thinking
A = lim An
n →∞
(
A = lim RRAMn 2 + x 2
n →∞
)

A = lim  2( b − a) + a2 ⋅ ( b − a) +
n →∞ 

 n + 1
 1  ( n + 1) ( 2n + 1) 
2
3
a
⋅
b
−
a
+
⋅
b
−
a
 (
)  6 
) 
 (
n⋅n
 n 
 


 ( n + 1) ( 2n + 1) 
 n + 1  1 
2
3
A = 2( b − a) + a2 ⋅ ( b − a) + a ⋅ ( b − a) ⋅ lim 

 +   ⋅ ( b − a) ⋅ lim 
n⋅n
 6

n →∞  n 
n →∞ 
 1
2
3
A = 2( b − a) + a2 ⋅ ( b − a) + a ⋅ ( b − a) ⋅ (1) +   ⋅ ( b − a) ⋅ (1) ⋅ ( 2)
 6
Applying this formula to the interval a = −3 to b = 2 yields:
€
1
2
3
2
A = 2( 2 − ( −3) ) + ( −3) ⋅ ( 2 − ( −3) ) + ( −3) ⋅ ( 2 − ( −3) ) ⋅ (1) +   ⋅ ( 2 − ( −3) ) ⋅ (1) ⋅ ( 2)
 6
€
€
 1
A = 2( 5) + 9( 5) + ( −3) ( 25) +   (125) ( 2)
 6
125
A = 10 + 45 − 75 +
3
65
A=
3
Comparing this to the definite integral:
€
(
)
2
A = ∫ −3
2 + x 2 dx
2

3 
x

A =  2x+


3

 −3
3


−3) 
23  
(

A =  2( 2) +
 −  2( −3) +

3
3

 

8
A = 4+ + 6+ 9
3
65
A=
3
€
Shelly Berman
p. 4 of 7
Area Approx Methods.doc
Jo Ann Fricker
Using Geometer’s Sketchpad to Support Mathematical Thinking
LRAM – Left Endpoint Rectangle Approximation Method
12
f(x ) = A⋅x 3 +B⋅x 2 +C⋅x+D
A = 0.00
A
Color Controls
Set Interval
10
B = 1.00
C = 0.00
B
8
C
D = 2.00
Area left
D
= 25.00
6
Animate n
4
n=5
x A = -3.00
x B = 2.00
h = 1.00
-8
2
-6
-4
Reset X-axis X-control
Reset Y-axis Y-control
A
B
-2
2
4
-2
Intermediate Information
-4
For the function f ( x ) = 2 + x 2 , the area under the curve using the left endpoint
2 − ( −3)
for n = 5 rectangles on the interval a = −3 to b = 2 and Δx =
= 1 can be
5
approximated by:
€
A5 = LRAM€5 f ( x )
€
5
€
€
= ∑ f ( xi−1 ) ⋅ Δx
i=1
= f ( −3) ⋅ 1 + f ( −2) ⋅ 1 + f ( −1) ⋅ 1 + f ( 0) ⋅ 1 + f (1) ⋅ 1
= 11 + 6 + 3 + 2 + 3
= 25
The generic formula for any approximation method and its limit can be derived
similarly.€
Shelly Berman
p. 5 of 7
Area Approx Methods.doc
Jo Ann Fricker
Using Geometer’s Sketchpad to Support Mathematical Thinking
MRAM – Midpoint Rectangle Approximation Method
12
f(x ) = A⋅x 3 +B⋅x 2 +C⋅x+D
A = 0.00
B = 1.00
A
10
B
C = 0.00
D = 2.00
Color Controls
Set Interval
8
C
D
6
Area midpoint = 21.25
Animate n
4
n=5
x A = -3.00
x B = 2.00
h = 1.00
-8
2
-6
-4
Reset X-axis X-control
Reset Y-axis Y-control
A
B
-2
2
4
-2
Intermediate Information
-4
Approximating the same area with five rectangles evaluated at the midpoint of
each subinterval:
A5 = MRAM5 f ( x )
5
 x + xi 
= ∑ f  i−1
 ⋅ Δx
2

i=1 
 5
 3
 1
1
 3
= f −  ⋅ 1 + f −  ⋅ 1 + f −  ⋅ 1 + f   ⋅ 1 + f   ⋅ 1
 2
 2
 2
 2
 2
33 17 9 9 17
=
+
+ + +
4
4 4 4 4
85
=
4
€
Shelly Berman
p. 6 of 7
Area Approx Methods.doc
Jo Ann Fricker
Using Geometer’s Sketchpad to Support Mathematical Thinking
TAM – Trapezoid Approximation Method
12
f(x ) = A⋅x 3 +B⋅x 2 +C⋅x+D
A = 0.00
B = 1.00
A
10
B
C = 0.00
D = 2.00
Color Controls
Set Interval
8
C
D
6
Area trapezoid = 22.50
Animate n
4
n=5
x A = -3.00
x B = 2.00
h = 1.00
-8
-6
Reset X-axis X-control
Reset Y-axis Y-control
2
-4
A
B
-2
2
4
-2
Intermediate Information
-4
Approximating the same area with five trapezoids:
A5 = TAM5 f ( x )
5
1
( f ( xi−1) + f ( xi )) ⋅ Δx
i=1 2
1
1
1
= ( f ( −3) + f ( −2) ) ⋅ 1 + ( f ( −2) + f ( −1) ) ⋅ 1 + ( f ( −1) + f ( 0) ) ⋅ 1
2
2
2
1
1
+ ( f ( 0) + f (1) ) ⋅ 1 + ( f (1) + f ( 2) ) ⋅ 1
2
2
1
1
1
1
1
= (11 + 6) + ( 6 + 3) + ( 3 + 2) + ( 2 + 3) + ( 3 + 6)
2
2
2
2
2
17 9 5 5 9
=
+ + + +
2 2 2 2 2
45
=
2
= ∑
€
Shelly Berman
p. 7 of 7
Area Approx Methods.doc
Jo Ann Fricker