Math131 Calculus I Areas & Distances Notes 5.1
I. Area
Under the Curve
Math131 Calculus I Notes 5.1 page
2
II. DEFINITION: The area A of the region S that lies under the graph of the continuous function f is
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III. SIGMA (SUMMATION) NOTATION:
=
ex#1 
ex#2 
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Thus the area under a curve can be written as:
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ex #3 Let A be the area of the region that lies
under the graph of 
between x = 0 and x =
2.
(a) Estimate the area by using four subintervals and the midpoints of the intervals.
(b) Improve your estimate of the area in part (a) by using 10 subintervals and the midpoints.
Math131 Calculus I Notes
5.1 page
3
ex #4 Suppose the odometer on our car is broken and we want to estimate the distance driven over a
30-second time interval. We take speedometer readings every five seconds and record them in the following table.
|
Time (s) |
0 |
5 |
10 |
15 |
20 |
25 |
30 |
|
Velocity (ft/s) |
25 |
31 |
35 |
43 |
47 |
46 |
41 |
(a) What would the calculation for distance traveled per five second interval look like?
What would the calculation for total distance look like?
(b) Draw a sketch of the line graph
that represents this data. Use time
on the horizontal axis and velocity
on the vertical axis.
(c) How are the calculations in part (a) related to the line graph in part (b)?
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In general…suppose an object moves with velocity v = f(t) where a < t < b and f(t) >0. Then…
Distance traveled during one
time interval =
Total distance traveled
during the interval [a,b] =
Thus, we conclude that the
distance traveled is equal to