Transcript Document

7.1 Integration by Parts
Integration by parts is an integration technique that
comes from the product rule for derivatives.
To simplify things while we introduce integration by parts. If
u is a function, denote its derivative by D(u) and an
antiderivative by I(u). Thus, for example, if u = 2x2, then
D(u) = 4x
and
I(u) =
[If we wished, we could instead take I(u) =
+ 46, but we
usually opt to take the simplest antiderivative.]
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Integration by Parts
Integration by parts
If u and v are continuous functions of x, and u has a
continuous derivative, then
Quick Example
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Example: Integration by Parts (Tabular Method)
Calculate:
Choose 1 function to be “u” and the other to be “v”. It is helpful to let “u”
equal the easiest function to take the derivative of.
Let u = x
Let v = ex
Use a table to calculate D(u) and I(v)
The table is read as
+x · ex −∫1 · ex dx
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Example: Repeated Integration by Parts
 x2e-x dx
Solve:
D
I
+
X2
e-x
-
2x
-e-x
= x2(-e-x) -  2x (-e-x) dx
The last integral is still a product.
Continue the table alternating
signs on the left.
D
I
+
X2
e-x
-
2x
-e-x
+
2
e-x
-
0
-e-x
= x2(-e-x) - 2x(e-x) + 2 (-e-x ) +C
= -x2e-x - 2xe-x -2e-x + C
= -e-x (x2 + 2x + 2) + C
To Summarize: Integrating a Polynomial Times a Function
If one of the factors in the integrand is a polynomial and the other factor is a function
that can be integrated repeatedly, put the polynomial in the D column and keep
differentiating until you get zero. Then complete the I column to the same depth, and
read off the answer.
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7.2 Area Between Two Curves and Applications
Area Between Two Graphs
If f (x) ≥ g(x) for all x in [a, b] (so that
the graph of f does not move below
that of g), then the area of the region
between the graphs of f and g and
between x = a and x = b is given by
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Example1: Area Between Two Curves
Find the area between f(x) = –x2 – 3x + 4 and
g(x) = x2 – 3x – 4
between x = –1 and x = 1
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Example2: Area Between 2 Curves
Find the Area between f (x) = | x | and g(x) = –| x – 1| over [–1, 2]
Remember:
 |x| dx = x |x| + C
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General Process: Area Between Two Curves
Finding the Area Between the Graphs of f (x) and g(x)
1. Find all points of intersection by solving f (x) = g(x) for x.
This either determines the interval over which you will
integrate or breaks up a given interval into regions
between the intersection points.
2. Determine the area of each region you found by
integrating the difference of the larger and the smaller
function. (If you accidentally take the smaller minus the
larger, the integral will give the negative of the area, so
just take the absolute value.)
3. Add together the areas you found in step 2 to get the
total area.
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7.3 Averages
To find the average of 20 numbers, add them up and divide
by 20. More generally, the average, or mean, of the n
numbers y1, y2, y3, . . . yn, is the sum of the numbers
divided by n. We write this average as (“y-bar”).
Average, or Mean, of a Collection of Values
The average of {0, 2, –1, 5} is
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Example: Average Speed
Over the course of 2 hours, my speed varied from 50 miles per hour to 60
miles per hour, following the function v(t) = 50 + 2.5t 2, 0 ≤ t ≤ 2.
What was my average speed over those two hours?
Recall : Average speed is total distance traveled divided by the time it took
and we can find the distance traveled by integrating the speed:
Distance traveled
It took 2 hours to travel this distance, so the average speed was
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Averages
The average, or mean, of a function f (x) on an interval [a, b] is
The average of f(x) = x on [1, 5] is
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Interpreting the Average of a Function Geometrically
Compare the graph of y = f(x) with the graph of
y = 3, both over the interval [1, 5]
We can find the area under the graph
of f(x) = x by geometry or by calculus;
it is 12. The area in the rectangle under
y = 3 is also 12.
Figure 8
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Interpreting the Average of a Function Geometrically
In general, the average of a positive function over the
interval [a, b] gives the height of the rectangle over the
interval [a, b] that has the same area as the area under
the graph of f(x)
The equality of these areas follows
from the equation
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Example: Average Balance
A savings account at the People’s Credit Union pays 3% interest,
compounded continuously, and at the end of the year you get a bonus
of 1% of the average balance in the account during the year. If you
deposit $10,000 at the beginning of the year, how much interest and
how large a bonus will you get?
Use the continuous compound interest formula A= Pert to calculate
the amount of money you have in the account at time t (in years) :
A(t) = 10,000e 0.03t
A(1) = $10,304.55 [Amount in account at end of 1 year]
So you will have earned $304.55 interest.
To compute the bonus: find the average amount in the account, which
is the average of A(t) over the interval [0, 1].
The bonus is 1% of this, or $101.52
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