1. Ch. 6 – The Definite Integral
6.1 – Estimating with Finite Sums

2. Ex: A car is traveling at a speed of 20 m/s from time t = 2 s to time t = 4 s. How much distance does the car travel from t = 2 to t = 4?
When velocity is constant, this is easy…use v = d/t to get d = 40 m.
How can we use a graph of velocity vs. time to find the answer to the example above?
Since v times t is distance, find the area of the rectangle under the line over the correct interval!
Secondary question: in terms of
calculus, how are d(t) and v(t) related?
v is the derivative of d!
Using the example at right, how do
we find the antiderivative of a function
using its graph?
Find the area under the curve!
v(t) (m/s)
t (s) 2 20
40 m
20 m/s
2 s 4

3. Given a graph of f(x),
Find the derivative of f by calculating the slope of f at every point (x, f(x)).
Find the antiderivative of f by calculating the area between f and the x-axis between 0 and x.
If f is not a straight line, this can be difficult because no geometric area formula may be useful:
To find this area, we will add together the areas of a whole bunch of skinny rectangles!
f(x) x1 x x2

4. Width of rectangles is the same for each rectangle!!!!
LRAM = Left-Hand Rectangular Approximation Method
RRAM = Right-Hand R.A.M.
MRAM = Midpoint R.A.M.
Height of rectangles intersect curve in the top left corner
Width of rectangles is the same for each rectangle!!!!
Height of rectangles intersect curve in the top right corner
Height of rectangles intersect curve in the midpoint of the top

5. Ex : Estimate the area of the region bounded by the graph of f(x) = 9 – x2 and the x- and y-axes using LRAM and 6 subintervals.
Each width is 3/6 = 0.5
How do we find the heights of the rectangles?
Use f(0), f(0.5), f(1), …, f(2.5)
for the height of the rectangles…
LRAM = (0.5)(9) + (0.5)(8.75) + …
+ (0.5)(5) + (0.5)(2.75)
Don’t use f(3) because it’s not a
left-hand value for any of the rectangles
LRAM = (0.5) ( … )
LRAM = x
f(x) 9
0.5
8.75 1 8
1.5
6.75 2 5
2.5
2.75 3
x = 3

6. Ex (cont.): Estimate the area of the region bounded by the graph of f(x) = 9 – x2 and the x- and y-axes using RRAM and 6 subintervals.
Each width is 3/6 = 0.5
How do we find the heights of the rectangles?
Use f(0.5), f(1), …, f(2.5), f(3)
for the height of the rectangles…
RRAM = (0.5)(8.75) + (0.5)(8) + …
+ (0.5)(2.75) + (0.5)(0)
Don’t use f(0) because it’s not a
right-hand value for any of the rectangles
RRAM = (0.5) ( … )
RRAM =
For MRAM, you’d use f(0.25),
f(0.75), …, f(2.25), f(2.75)
for the heights x
f(x) 9
0.5
8.75 1 8
1.5
6.75 2 5
2.5
2.75 3
x = 3

7. Ex: Use a calculator to estimate the area of the region bounded by the graph of f(x) = 2 + xsinx over [0, 7π/6] using the following RAM and subintervals:
Actual area is about square units
Notice that as n increases, the RAMs become almost identical, and they will be increasingly accurate.
n (# subintervals)
LRAMn
MRAMn
RRAMn 2 10
100
1000