7.3 Rectangular Approximation Method
Up to this point we’ve dealt with differential calculus (involving the concept of tangents
to curves and derivatives). We will now move on to integral calculus, which deals
primarily with areas under curves.
Example: If I travel at 75 mph (constant velocity) between 10 am and noon how far did I
travel?
velocity
75
This area represents
distance traveled
10
12
time
But what if the velocity was not constant?
velocity
The area under the curve
is still distance traveled,
but since the area is not
rectangular it is difficult
to find.
10
12
Time
7.3 Rectangular Approximation Method
Rectangular Approximation:
Example: A particle starts at x = 0 and moves along the x-axis with velocity v(t )  t 2 for
t  0 . Where is the particle at t = 3?



















We can estimate the area
under the curve by
summing up the areas of
small rectangular strips.
The more strips the better
the approximation.







RAM – Rectangular Approximation Method:
Steps: 1) Divide the interval in question into n subintervals
2) Find the midpoint (MRAM), left endpoint (LRAM) or right endpoint
(RRAM) of each region, depending on which specific method you want to
use.
3) The height of the rectangle in each subinterval will be the value of the
function at either the midpoint (MRAM) left endpoint (LRAM) or right
endpoint (RRAM).
length of interval
4) The width of each rectangle is
n
7.3 Rectangular Approximation Method
MRAM n = 3
Midpoint Rectangular
Approximation Method
Area of Rectangle 1 =
(.5) 2  1  0.25



Area of Rectangle 2 =
(1.5) 2  1  2.25



Area of Rectangle 3 =
(2.5) 2  1  6.25


TOTAL AREA = 8.75











RRAM n =3
Right endpoint Rectangular
Approximation Method
Area of Rectangle 1 =
(1) 2  1  1



Area of Rectangle 2 =
(2) 2  1  4



Area of Rectangle 3 =
(3) 2  1  9


TOTAL AREA = 14











LRAM n = 3
Left-endpoint Rectangular
Approximation Method
Area of Rectangle 1 =
(0) 2  1  0



Area of Rectangle 2 =
(1) 2  1  1



Area of Rectangle 3 =
(2) 2  1  4


TOTAL AREA = 5











7.3 Rectangular Approximation Method
Example 1:
Given f ( x)  2 x .
Approximate the area of the region under the graph and above the x axis between x = 0
and x = 4 by using four rectangles of equal width.
Use the LRAM, R Ram, and MRAM methods.
The Definite Integral
Suppose f is defined on the interval [a,b], the definite integral of f from a to b
b
is given by

n
f ( x)dx  lim  f ( xi )x
n 
a
i 1
provided the limit exists, where x 
(b  a )
n
Our goal is to approximate the total accumulated change from the rate of change.
and to estimate the distance traveled when the velocity is not constant. In other words,
given the velocity of an object, find the total distance traveled.
Example 2:
A driver traveling on a business trip checks the speedometer every hour. The table
shows the driver’s velocity at several times.
Time (hour)
Velocity (mph)
0
0
1
52
2
58
3
60
Approximate the total distance traveled during the three hour period.
Section 7.3 # 5, 7, 9, 25,39