Indefinite Integrals and the Net Change

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Indefinite Integrals and the Net Change

• Indefinite Integrals
• Table of Indefinite Integrals
• Net Change of a Function
• Distance Traveled
• Which Car Wins

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Indefinite Integrals
Definition
An antiderivative F of a function f is called
an indefinite integral of f.
Notation
Example
since
Remark
Most computer mathematics systems abuse this
notation for practical reasons and interpret it
as a specific antiderivative rather than the
family of all antiderivatives.
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Table of Indefinite Integrals
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Computing Indefinite Integrals
Example 1
Solution
This integral can be computed using the table
repeatedly.
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Computing Indefinite Integrals
Example 2
Solution
To compute this integral one needs to rewrite the
function to be integrated so that the integral
can be computed by the table of integrals.
To get
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Net Change
By the Fundamental Theorem of Calculus we have,
for an indefinite integral function F of f
This can be rewritten as follows
Definition
The quantity F(b) F(a) is the net change of
the function F over the interval a,b. The
derivative F(x) is the rate of change of the
function F.
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Distance Traveled
The picture shows the speed (in meters/second) of
two race cars during the first 20 seconds of the
race.
Problem
Which car is ahead after 20 seconds of the race?
In this problem the functions, i.e., the speeds
of the cars, are defined by the tachometer and
not by a mathematical expression.
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Estimating Distance Traveled (1)
To estimate the distance traveled by the car A
whose speed is given, one has to divide the time
interval from 0 to 20 seconds into subintervals
in which the speed can be taken to be constant.
Clearly, for an accurate estimate, one needs a
large number of time subintervals.
The distance the car has traveled during the
first 2 seconds ? (speed at time 1 second)(2
seconds)
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Estimating Distance Traveled (2)
Let
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Estimating Distance Traveled (3)
As n grows, the sum estimating the distance
traveled approaches the integral of the speed
over the time interval in question.
Conclude
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Estimating Distance Traveled (4)
The speed s(t) at time t is not given by a
mathematical function but by the tachometer.
Problem
We have no mathematical method to compute an
antiderivative of a tachometer. An odometer in a
car is an antiderivative of the tachometer, but
we do not have odometer readings. The only data
available is the tachometer data.
So the integral giving the distance traveled
cannot be computed by the Fundamental Theorem of
Calculus.
We have to use
The larger n is, the better the estimate.
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Which Car Wins?
From the data given by the tachometer graph one
can compute Riemann Sum estimates for the
distances the two cars have covered during the
first 20 seconds.
Numerical Estimates
Car A
Here we use n40.
Car B
Winning car
Speed is given in meters/second, time in seconds.
Hence the unit for the distance traveled is
(meters/second) second meters