Trapezoidal Approximation
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Transcript Trapezoidal Approximation
Trapezoidal Approximation
Objective: To relate the Riemann
Sum approximation with rectangles
to a Riemann Sum with trapezoids.
Trapezoidal Approximation
• We will now look at finding area by using trapezoids
rather than rectangles. Remember, the area of a
h
trapezoid is A ( b b ) . In the picture below, the
2
height of each trapezoid is x . The bases of each
trapezoid are the parallel sides. For us, this will be
the value of the function
evaluate at the endpoints
of each “strip.”
1
2
Trapezoidal Approximation
• For example, the area of the first trapezoid would be:
height x
A
2
A1
1
n
2
b 2 y1
b1 y 0
h
ba
( b1 b 2 )
ba
2n
( y 0 y1 )
height
1
2
x
ba
2n
( b1 b2 ) ( y 0 y1 )
Trapezoidal Approximation
• For example, the area of the second trapezoid would
be:
A1
A2
ba
2n
ba
2n
( y 0 y1 )
( y1 y 2 )
Trapezoidal Approximation
• For example, the area of the third trapezoid would
be:
A1
A2
A3
ba
2n
ba
2n
ba
2n
( y 0 y1 )
( y1 y 2 )
( y2 y3 )
Trapezoidal Approximation
• For example, the area of the nth trapezoid would be:
A1
A2
A3
An
ba
2n
ba
2n
ba
2n
ba
2n
( y 0 y1 )
( y1 y 2 )
( y2 y3 )
( y n 1 y n )
Trapezoidal Approximation
• When adding these areas together and factoring out
the common b a it becomes
A1
A3
A
ba
2n
ba
2n
ba
2n
2n
( y 0 y1 )
( y2 y3 )
A2
An
ba
2n
ba
2n
( y1 y 2 )
( y n 1 y n )
( y 0 y1 y1 y 2 y 2 y 3 ... y n 1 y n )
Trapezoidal Approximation
• Notice how each value of y is used twice except for
the first and last ones. This leads us to another form
of the equation.
A
ba
2n
A
ba
2n
( y 0 y1 y1 y 2 y 2 y 3 ... y n 1 y n )
( y 0 2 y1 2 y 2 2 y 3 ... 2 y n 1 y n )
f ( x ) dx T n
ba
2n
( y 0 2 y1 2 y 2 2 y 3 ... 2 y n 1 y n )
Example
• Selected values of a continuous function are given in
the table. Using 10 subintervals of equal length, the
Trapezoidal Rule approximation for f ( x ) dx is
10
0
Example
• Selected values of a continuous function are given in
the table. Using 10 subintervals of equal length, the
Trapezoidal Rule approximation for f ( x ) dx is
10
0
ba
2n
10
0
f ( x ) dx
10 0
2 (10 )
1
2
1
2
20 2 (19 . 5 ) 2 (18 ) 2 (15 . 5 ) 2 (12 ) 2 ( 7 . 5 ) 2 ( 2 ) 2 ( 4 . 5 ) 2 ( 12 ) 2 ( 20 . 5 ) ( 30 )
10
0
f ( x ) dx 32 . 5
Example
• Using the subintervals [1, 5], [5, 8], and [8, 10], what
is the trapezoidal approximation to
?
( 2 cos x ) dx
10
1
a )17 . 126
b )17 . 129
c )17 . 155
d )18 . 147
e )19 . 386
Example
• Using the subintervals [1, 5], [5, 8], and [8, 10], what
is the trapezoidal approximation to
?
( 2 cos x ) dx
10
1
• Trap 1
5 1
2
[( 2 cos 1) ( 2 cos 5 )]
6 . 352
• Trap 2
• Trap 3
85
2
10 8
2
[( 2 cos 5 ) ( 2 cos 8 )]
a )17 . 126
b )17 . 129
c )17 . 155
5 . 7927
d )18 . 147
[( 2 cos 8 ) ( 2 cos 10 )]
e )19 . 386
4 . 9846
Example
• Using the subintervals [1, 5], [5, 8], and [8, 10], what
is the trapezoidal approximation to
?
( 2 cos x ) dx
10
1
• Trap 1
5 1
2
[( 2 cos 1) ( 2 cos 5 )]
a )17 . 126
b )17 . 129
• Trap 2
85
2
[( 2 cos 5 ) ( 2 cos 8 )]
c )17 . 155
d )18 . 147
• Trap 3
10 8
2
[( 2 cos 8 ) ( 2 cos 10 )]
e )19 . 386
Example
• If three equal subdivisions of [0, 3] are used, what is
the Trapezoidal Rule approximation of ( x 6 x 9 ) dx ?
3
2
0
a )3
b )9
c )9 .5
d )10
e )19
Example
• If three equal subdivisions of [0, 3] are used, what is
the Trapezoidal Rule approximation of ( x 6 x 9 ) dx ?
3
2
0
• Trap 1
1
2
[9 4 ] 6 .5
a )3
b )9
• Trap 2
1
2
[ 4 1] 2 . 5
c )9 .5
d )10
• Trap 3
1
2
[1 0 ] . 5
9 .5
e )19