AP Calculus Notes Chapter 4 Section I
1/18/12-1/20/12
Approximating Definite Integrals Using Riemann Sums and Trapezoidal Sums
Riemann Sums (Rectangles)
Right Riemann Sum
The graph of the function f is shown below. Approximate
 f  x dx using a right Riemann sum with 3
4
2
subintervals of equal length. Sketch the graph of the right Riemann sum.

y




x






Left Riemann Sum
The graph of the function f is shown below. Approximate
 f  x dx using a left Riemann sum with 3
4
2
subintervals of equal length. Sketch the graph of the left Riemann sum.

y




x






Midpoint Riemann Sum
The graph of the function f is shown below. Approximate
 f  x dx using a midpoint Riemann sum with
4
2
3 subintervals of equal length. Sketch the graph of the midpoint Riemann sum.

y




x






Trapezoidal Sums
Trapezoidal Sum
The graph of the function f is shown below. Approximate
 f  x dx using a trapezoidal sum with 3
4
2
subintervals of equal length. Sketch the graph of the trapezoidal sum.

y




x






y
y
y
x
The left Riemann sum ___
 g  x dx
b
x
The right Riemann sum ___
a
y
 g  x dx
b
a
 g  x dx
b
The right Riemann sum ___
a
a
x
 g  x dx
b
 g  x dx
b
The right Riemann sum ___
a
 g  x dx
b
x
The trapezoidal sum ___
a
x
b
b
y
y
 g  x dx
 g  x dx
x
a
y
The trapezoidal sum ___
a
y
a
b
y
x
The left Riemann sum ___
 g  x dx
x
y
The left Riemann sum ___
The trapezoidal sum ___
y
x
The left Riemann sum ___
x
 g  x dx
b
a
y
x
The right Riemann sum ___
x
 g  x dx
b
The trapezoidal sum ___
a
The left Riemann Sum overapproximates the value of definite integral when _______ .
The left Riemann Sum underapproximates the value of definite integral when _______ .
The right Riemann Sum overapproximates the value of definite integral when _______ .
The right Riemann Sum underapproximates the value of definite integral when _______ .
The trapezoidal sum overapproximates the value of definite integral when _______ .
The trapezoidal sum underapproximates the value of definite integral when _______ .
 g  x dx
b
a
A school carnival sold raffle tickets as a fundraiser between noon (t = 0) and 8 p.m. (t = 8). The
number of raffle tickets sold t hours after noon is modeled by a differentiable function R for
0  t  8 . R  t   0 for all t.
Values of R  t  , in hundreds of tickets, at various time t are shown in the table below.
t
(hours)
R t 
(hundreds of tickets)
0
2
5
6
8
0
5
9
10
11
a) Use a trapezoidal sum with the four subintervals given by the table to approximate
Does this approximation overestimate or underestimate the actual value of
 R t  dt
8
0
 R t  dt ?
8
0
Give a reason for your answer.
b) Use a right Riemann sum with the four subintervals given by the table to approximate
Using correct units, explain the meaning of
1 8
R  t  dt
8 0
1 8
R  t  dt , in terms of the number of raffle tickets.
8 0
During the time interval 0  t  12 hours, water is pumped into a swimming pool at the rate
P  t  cubic feet per hour. The table below gives values of P  t  , a strictly monotonic differentiable
function, for selected values of t.
t
0
2
4
6
8
10
12
P t 
0
12
20
25
30
50
55
a) Using correct units, explain the meaning of

12
0
P  t dt and
1 12
P  t dt in terms of the water in the
12 0
swimming pool.
b) Use a midpoint Riemann sum with three subintervals of equal length to approximate

12
0
P  t dt .
Show the computations that lead to your answer.
1 12
P  t dt .
12 0
Does this approximation overapproximate or
c) Use a left Riemann sum with six subintervals of equal length to approximate
Show the computations that lead to your answer.
1 12
P  t dt ? Justify your answer.
underapproximate the actual value of
12 0
Homework – Due Monday 1/23/12
Problem Set 48
Handout Problems #1-10 all