Name: ________________________ Period: ______ Sec 4.6 Notes (Trapezoid Rule)
Assignment: Problems in this handout and (see instructions at the bottom of PAGE 2)
Sec4-5: #76-86 evens, 87, 89 and Sec4-6: # 39 & 52 (Trapezoid Rule Only)
The Fundamental Theorem of Calculus gives us a way to find the exact value for definite
integrals by using anti-derivatives. However, there are many functions that simply do not have
anti-derivates that are elementary functions. For definite integrals involving these functions,
the Fundamental Theorem is not useful in finding the exact value. Today’s lesson involves a
method to approximate definite integrals when the exact value can not be found using the
Fundamental Theorem. The area of trapezoids will be used to approximate definite integrals.
Using trapezoids to approximate these definite integrals allows a method to calculate the
amount of error associated with the approximate value. The Trapezoidal estimate is very
similar to a Riemann Sum estimate, except we use trapezoids rather than rectangles and we
have a method for calculating the maximum error.
Example: The Fundamental Theorem is not useful to help evaluate these five definite
integrals because their integrand doesn’t have an anti-derivative that is an elementary
function.


0
 x
3
1
10

1  x dx ,

1  x 3 dx ,
-7

2
1
1.7
0.5
x cos( x) dx ,

0.5
0.1
cos( x)
dx
x
sin( x 2 ) dx
Shaded area represents the trapezoid
Area of Trapezoid:
approximation of
1
A  (b1  b 2)h
2

b
a
y4
f ( x) dx
y3
y0
y1
y5
b1
y6
b2
y2
h
h  x 
ba
n
a
b
The trapezoid estimate of this definite integral is shown. SIMPLIFY this formula by factoring.
1
1
1
1
1
1
T6  ( y0  y1 )x  ( y1  y 2 )x  ( y 2  y3 )x  ( y3  y 4 )x  ( y 4  y5 )x  ( y5  y6 )x
2
2
2
2
2
2
Simplified Trapezoid Rule to
approximate

b
f ( x) dx
a
y 
y
T6  x 0  y1  y 2  y3  y 4  y5  6 
2 
 2
Example: Find the exact value by using the Fundamental Theorem of Calculus.
Approximate using the Trapezoid Rule with 8 trapezoids (show work)
Calculate the error in the trapezoid estimate.
Fundamental Theorem


0
sin(x) dx
Trapezoid Estimate
a. Calculate x 
ba
& draw trapezoids on graph.
n
b. Fill in the table with sample points
and the y-values you need to use
in the trapezoid formula
.
x
y
y0 =
y1 =
y2 =
y3 =
y4 =
y5 =
y6 =
y7 =
y8 =
c.
T6  ______________________________________________
Calculate the error
Calculation of the maximum error made when using the Trapezoid Rule.
PAGE 2
When using the Trapezoid Rule to approximate a definite integral it is important to know how
accurate you can expect the approximation to be.
In the previous example, we were able to calculate the error when we used the Trapezoid Rule
because we were able to find the exact value of the definite integral using the Fundamental
Theorem.
Normally, you would only use the Trapezoid Rule if you can NOT find the exact value by using
the Fundamental Theorem. In these cases, you can use the rule below to calculate the
maximum error. Note that the error may be smaller than the maximum and this formula may
only be applied when the second derivative is continuous over the limit of integration.
Trapezoid Maximum Error Formula:
Tn  
b
a
f ( x) dx
with a maximum error equal to E.
(b  a)3

E 
max f ( x)  , a  x  b
2
12n
Example: Calculate the maximum error made with a Trapezoid estimate of the definite
integral using n=8 trapezoids. How does this maximum error compare to the actual
error you calculated previously?


0
sin(x) dx
Assignment: Approximate all the integrals listed on the first page using the Trapezoid Rule
with n = 4 trapezoids. Calculate the maximum error on one of the integrals.





Sketch the graph showing all trapezoids
Fill in table with sample points and y-value
Show calculation of Trapezoid estimate
On one problem, show work to estimate the maximum error.
Explain how the concavity affects the trapezoid estimate (over or under estimates?)
(1) Estimate

0
3 x
1
1  x dx
using four trapezoids.
Sketch a graph showing the trapezoids.
x 
ba
= ________________
n
Fill in the table.
x
y
y0 =
y1 =
y2 =
y3 =
y4 =
T4  ______________________________________________
Concavity on the interval? ______________________ Over or Under Estimate?
(2) Estimate


0
2
x cos( x) dx using four trapezoids.
Sketch a graph showing the trapezoids.
x 
ba
= ________________
n
Fill in the table.
x
y
y0 =
y1 =
y2 =
y3 =
y4 =
T4  __________ __________ __________ __________ ______
Concavity on the interval? ______________________ Over or Under Estimate?
(3) Estimate
.

0.5
0.1
cos( x)
dx using four trapezoids. Sketch a graph showing the trapezoids.
x
ba
= ________________
n
x 
Fill in the table.
x
y
y0 =
y1 =
y2 =
y3 =
y4 =
T4  ______________________________________________
Concavity on the interval? ______________________ Over or Under Estimate?
(4) Estimate
x 

1
-7
1  x 3 dx using four trapezoids. Sketch a graph showing the trapezoids.
ba
= ________________
n
Fill in the table.
x
y
y0 =
y1 =
y2 =
y3 =
y4 =
T4  ______________________________________________
Concavity on the interval? ______________________ Over or Under Estimate?
(5) Estimate
x 

1.7
0.5
sin( x 2 ) dx using four trapezoids. Sketch a graph showing the trapezoids.
ba
= ________________
n
Fill in the table.
x
y
y0 =
y1 =
y2 =
y3 =
y4 =
T4  ______________________________________________
Concavity on the interval? ______________________ Over or Under Estimate?
(6) Write down the interval you are selecting to calculate the error made by using a trapezoid
estimate with n = 4 rectangles.
Hint. You need to find the second derivative. Then, graph the absolute value of the second
derivative. Look at that graph over the limit of integration. Find the maximum y-value.
Then, use that maximum y-value in the calculation of the maximum error.
Show all work.