AP Calc Notes: DI – 2 Riemann Sums Applications
Problem: Approximate the area under the
curve y = f(x) from x = a to x =
b.
y
y = f(x)
1. Partition the interval [a, b] into n
subintervals. Let xi be the right
endpoint of the ith subinterval.
Note that it is not necessary that the
subintervals all be the same width.
The width of the ith interval is
Δxi = xi - xi - 1 = xright − xleft
2.
A2
A1
a
x0
x1
A3 . . . . .
x2 x3
Ai
......
xi-1 xi
An-1 An
xn-2 xn-1 b
Δxi
x
xn
Let xi* be any x-value in the ith subinterval: xi* could be the left endpoint, the right endpoint, the
midpoint, or any other point in [xi-1, xi]. Then the approximate area under the curve in the ith subinterval is
a rectangle having width Δx and height yi*.
Then for the ith rectangle
base = Δxi
Ai
height = f(xi* )
Ai ≈ bh = f(x )Δxi
*
i
xi-1 xi
xi*
3. Finally, an approximation to the area under the curve is the sum of the approximations to the areas under
the individual subintervals:
A ≈ A1 + A2 + . . . + An = f(x1* )Δx1 + f(x2* )Δx2 + f(x3* )Δx3 + . . . + f(xn* )Δxn
A≈
n
∑ f(x )Δx
i=1
*
i
i
This is called a Riemann sum (add area of rectangles).
Riemann sums typically come in three flavors: left, right and midpoint. All approximate area
under a curve. If the subintervals are all the same width (which is NOT a requirement),
b-a
.
then Δx =
n
Position, velocity and acceleration review
Position
m
s (t )
Velocity
m/s
s '(t )
v (t )
Acceleration
m/s2
s "(t )
v '(t )
a (t )
Given v ( t ) , find
Change in Position
Total Distance Traveled
v(t)
v(t)
1
1
t
1
2
3
4
5
t
6
1
−1
−1
−2
−2
2
3
4
5
6
Ex 1) Particle A moves along a horizontal line with a velocity, v ( t ) , where v ( t ) is a positive continuous
function of t. The time t is measured in seconds, and the velocity is measured in cm/sec. The velocity v ( t ) of
the particle at selected times is given in the table below.
t (sec)
0
2
5
7
10
v ( t ) (cm/sec)
1.7
6.8
7.4
15.6
24.9
Use data from the table to approximate the distance traveled by particle A over the interval 0 ≤ t ≤ 10 seconds
by using a right Riemann sum with four subintervals. Show the computations that lead to your answer, and
indicate units of measure.
Ex 2): The table below shows the velocity of a model train engine moving along a track for 10 min. Estimate
the distance traveled by the engine, using 5 subintervals of a left Riemann sum.
v ( t ) (in/s)
t (min)
0
0
12
1
22
2
10
3
5
4
13
5
11
6
6
7
2
8
6
9
0
10
Ex 3): (with calculator) Brad Pitstop is trying to set a new land speed record in his rocket powered car. After
six seconds, the rocket malfunctions and Brad brakes to a stop. His velocity in meters per second from
t 5/ 2
time t = 6 until he stops is given by v ( t ) =
− 4t 3/ 2 + 60t1/ 2 . Approximate the distance Brad travels after
15
the rocket malfunctions with a Riemann sum using left-hand Riemann approximations of 8 equal
subintervals.
v
Brad stops when v(t) = 0 → t = 30.
d=
n
∑ v Δt where Δt =
i=1
i
30 - 6
=3
8
t
6