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
ba
( b1  b 2 )
ba
2n
( y 0  y1 )
height 
1
2
x 
ba
2n
( b1  b2 )  ( y 0  y1 )
Trapezoidal Approximation
• For example, the area of the second trapezoid would
be:
A1 
A2 
ba
2n
ba
2n
( y 0  y1 )
( y1  y 2 )
Trapezoidal Approximation
• For example, the area of the third trapezoid would
be:
A1 
A2 
A3 
ba
2n
ba
2n
ba
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 
ba
2n
ba
2n
ba
2n
ba
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
ba
2n
ba
2n
ba
2n
2n
( y 0  y1 )
( y2  y3 )
A2 
An 
ba
2n
ba
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
ba
2n
A
ba
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 
ba
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
ba

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
85
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
85
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