MATH 131-503 Fall 2015
c
Wen
Liu
5.1
Chapter 5 Integrals
5.1 Areas and Distances
Definition: Given a continuous function f (x) on the interval [a, b]. We divide the interval into n
subintervals of equal length
b−a
4x =
n
The endpoints of the subintervals are
a = x0 , x1 , x2 , · · · , xn−1 , xn = b
We then define
Ln = f (x0 )4x + f (x1 )4x + · · · + f (xn−1 )4x =
n
X
f (xi−1 )4x
i=1
Rn = f (x1 )4x + f (x2 )4x + · · · + f (xn )4x =
n
X
f (xi )4x
i=1
We use the term left-hand sum Ln because we use the points at the left-hand ends of the subintervals and use the term right-hand sum Rn because we use the points at the right-hand ends of
the subintervals. We often use sigma notation to write sums with many terms more compactly.
Furthermore,
A =
=
lim Ln
n→∞
lim Rn
n→∞
where A is the area of the region S between the graph of f (x) and the x-axis on [a, b].
Note:
• It can be shown that we get the same value if we use any number x∗i , which is called the sample
points, in the ith subinterval [xi−1 , xi ]. Hence
A =
=
=
lim
n→∞
lim
n→∞
lim
n→∞
n
X
i=1
n
X
i=1
n
X
f (xi−1 )4x
f (xi )4x
f (x∗i )4x
i=1
• The more rectangles we use, the better we estimate.
Page 1 of 3
MATH 131-503 Fall 2015
5.1
c
Wen
Liu
Example 1: Estimate the area under the graph of y = x2 from 0 to 1.
(a) Use four approximating rectangles and left endpoints. Is your estimate an underestimate or an
overestimate?
(b) Repeat part (a) using right endpoints.
(c) Repeat part (a) using middle points.
Page 2 of 3
MATH 131-503 Fall 2015
5.1
c
Wen
Liu
Background: If our car travels at a constant velocity of 55miles per hour
for 2 hours, then, since distance is the product of velocity with time (d = vt),
we have traveled 55 · 2 = 110 miles. Let us represent this graphically. If we
let v = v(t), then the distance traveled is the area of the region between the
graph of v = v(t) = 55 and the t-axis on [0, 2]. But what happens if velocity
varies from instant to instant? How can we determine the distance traveled?
In general, the distance traveled is the area bounded between v(t) and the t-axis on [a, b].
Inspired by the examples above, we use rectangles to estimate the area.
Example 2: Speedometer readings for a motorcycle at 12-second intervals are given in the table.
t(s)
0
v(f t/s) 32
12 24 36 48 60
28 25 22 24 27
(a) Estimate the distance traveled by the motorcycle during this time period using the velocities at
the beginning of the time intervals. Is your estimate an underestimate or an overestimate?
(b) Give another estimate using the velocities at the end of the time periods. Is your estimate an
underestimate or an overestimate?
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