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2. Definite
Integrals and
Numeric
Integration
Calculus
• Calculus answers two very important
questions.
• The first, how to find the instantaneous
rate of change, we answered with our
study of derivatives
• The second we are now ready to
answer, how to find the area of
irregular regions.
Sigma Notation
• Remember sigma notation from precalculus?
• The sum of n terms a , a , a , ..., a is written as
1
2
3
n
n
S 
a
i
i 1
• The i tells you where to start and end
summing.
Approximating Area
• We will now approximate an irregular area
bounded by a function, the x-axis between
vertical lines x=a and x=b, like the one
below by finding the areas of many
rectangles and summing them up.
Finding area
• Break region into subintervals (strips)
• These strips resemble rectangles
• Sum of all the areas of these “rectangles”
will give the total area
Rectangular Approximation
Method (RAM)
• Since the height of the rectangle varies along the
subinterval, in order to find area of the rectangle, we
must use either the left hand endpoint (LRAM) to
find the height, the right hand endpoint (RRAM) or
the midpoint (MRAM)
• The more rectangles you make, the better the
approximation
Rectangular Approximation
Method (RAM)
• If a function is increasing, LRAM will underestimate
the area and RRAM will overestimate it.
• If a function is decreasing, LRAM will overestimate
the area and RRAM will underestimate it
Trapezoid Approximation
• Another approximation we can use (and probably the
best) is trapezoids.
• Trapezoids give an answer between the LRAM and
RRAM
• The formula for the area of a trapezoid is
½(x)(y1+y2)
Example 1
• A particle starts at x=0 and
moves along the x-axis with
velocity v(t)=t2. Where is the
particle at t=3? Use interval
width of ¼ and MRAM.
• Find height at each midpoint of
interval. Multiply height times
width to get area. Sum all the
areas.
Subinterval
[0, ¼] [¼, ½]
[½,3/4]
[3/4, 1]
Midpoint
1/8
3/8
5/8
7/8
Height (t2)
1/64
9/64
25/64
49/64
Area
1/256
9/256
25/256
49/256
• Add all
1/256+9/256+25/256+49/256
+81/256+121/256+169/256+
225/256+289/256+361/256+
441/256+529/256 = 8.98
etc
Example 2
• Find the area under y=x2 from x=0 to x=3,
use width of ½. Use all methods
• LRAM
0 (1 / 2)  (1 / 2) (1 / 2)  1 (1 / 2)  (3 / 2) (1 / 2)  2 (1 / 2)  (5 / 2) (1 / 2)
2
2
2
2
2
2
1 / 2[0  (1 / 2)  1  (3 / 2)  2  (5 / 2) ]
2
2
2
2
2
2
6 .8 7 5
• MRAM
1 / 2[(1 / 4)  (3 / 4)  (5 / 4)  (7 / 4)  (9 / 4)  (11 / 4) ]
2
• RRAM
2
2
2
2
2
1 / 2[(1 / 2)  (1)  (3 / 2)  (2)  (5 / 2)  (3) ]
2
2
2
2
2
2
8 .9 3 7 5
1 1 .3 7 5
• Trap
1 / 2(1 / 2)[(0  2(1 / 2)  2(1)  2(3 / 2)  2(2)  2(5 / 2)  (3) ]
2
2
2
2
2
2
2
9.125
Example 3
• Find the area under y=x2+2x-3 from x=0
to x=2, use width of ½
• LRAM
1 / 2[  3   1.75  0  2.25]
 1 .2 5
• MRAM
1 / 2[  2.4375   .9375  1.0625  3.5625]
• RRAM
1 / 2[  1.75  0  2.25  5]
.6 2 5
2 .7 5
• Trap
(1 / 2)(1 / 2)[  3  2(  1.75)  2(0)  2(2.25)  5]
0.75
How many rectangles
should we make?
• The estimate of area gets more and
more accurate as the number of
rectangles (n) gets larger
How many rectangles
should we make?
• If we take the limit as n approaches
infinity, we should get the exact area
• We will take more about this
tomorrow…..
Worksheet notation
L n  L R A M w ith n intervals
M n  M R A M w ith n intervals
R n  R R A M w ith n intervals
T n  T rapezoids w ith n intervals
If interval is 3 units long and you have 6 subintervals,
Each subinterval will be 3/6 or ½ wide.
BREAK!!
Remember from
yesterday…..
• We were talking about increasing the number of
rectangles giving us a better estimate of the area
• What if we took the limit as n approached
infinity??
• The area approximation would approach the
actual area
• The process of finding the sum of areas of
rectangles to approximate area of a region is
called a Riemann Sum, after Bernhard Riemann
Riemann Sums
• Riemann proved that the finite process of
adding up the rectangular areas could be found
by a process known as definite integration.
Here is the essence of his great, time-saving
work.
b
b
A  lim
n 

ia
f ( xi )  x 

a
f ( x ) dx
What the notation means
Upper limit of integration
b
Integral symbol

Integrand
f ( x ) dx
a
Variable of Integration
Lower limit of integration
Read this as integral from a to b of
f of x dx
Example 4
4
• Evaluate  2 xdx
geometrically as well as
0
on your calculator
Negative area?
• The example we just looked at was nonnegative on the interval we evaluated. This is
not always the case.
• If f(x) is non-negative and integratable over a
closed interval [a,b] then the area under the
curve is the definite integral of f from a to b
• If f(x) is negative and integrable over a closed
interval [a,b], then the area under the curve is
the OPPOSITE of the definite integral of f from
a to b.
b
A    f ( x ) dx
a
In general…
b
•

f ( x ) dx
does NOT give us area but rather
the NET accumulation over the interval x=a
to x=b. If f(x) is positive and negative on a
closed interval, then  f ( x ) dx will NOT give us
area.
a
b
a
• When integrating left to right, regions above the x-axis are
positive and regions below the x-axis are negative.
• When integrating right to left, regions above the x-axis are
negative and regions below the x-axis are positive.
• This can be summarized as
b

a
a
f ( x ) dx    f ( x ) dx
b
Negative functions
• When using definite integrals to find area,
you must divide the interval into
subintervals where function is positive and
where it is negative and use absolute
values of definite integral
• When using area to find definite integrals,
you must assign the correct sign to the
area.
Example 5
• The graph of f(x) is shown below. If A1 and A2 are
positive numbers that represent the areas of the
shaded regions, then find the following.
Another property of
definite integrals
• The property that allows us to
do the calculations in the
previous example is
b

a
c
f ( x ) dx 

a
b
f ( x ) dx 

c
f ( x ) dx
Example 6
• Approximate  ( x  4) dx
using four subintervals of
equal length and trapezoidal method. Test your
answer against the calculator’s approximation
using fnint. Can any of these approximations
represent the area of the region? Why or why not?
5
2
1
Example 7
• Find the area in the previous problem using
trapezoids and also set up integrals needed to
calculate with calculator.
Another way to find area
with calculator
b
A

a
f ( x ) dx
Example 8
• We can also find areas when our function is given to
us in either data form or graph form.
3

f ( x ) dx
• Approximate
using LRAM, MRAM, RRAM,
and trapezoids. Do these represent area? Also
approximate f’(1)
0
Example 9
8
Approximate  f ( x ) dx
using LRAM, RRAM,
and trapezoids. Do these represent area?
Why did I leave off MRAM? Also
approximate f’(7)
1
Example 10
• Sketch the region corresponding to each
definite integral, then evaluate each
integral using a geometric formula.
Decide if the integral represents the area.
Example 11
0

5
f ( x ) dx

f ( x ) dx
• Evaluate
and
. Do these
represent the area of the region? Why or
why not? If not, what is the area of the
region?
7
2
Properties of definite
integrals
• We have seen some of these
already.
Example 12
• Given that
1

4
f ( x ) dx  5,
1

1
f ( x ) dx   2,
 h ( x ) dx  7
1
1
• Find
1
1

f ( x ) dx

2
f ( x )dx
 can 't d o
 h ( x )dx
 can 't d o
4
4
0

2
f ( x ) dx
 3
1
2
1
 2 f ( x )  3 h ( x ) dx
1
 31
Example 13
8
10
• If

0
f ( x ) dx  17
and

0
f ( x ) dx  12
10
, find
 (3 f ( x )  2) dx
8
Example 14
 9  x2
 x3
5 
6
•

dx

using area of the region
to help evaluate the integral.
• If

 sin xdx  2
Example 15
use this fact and symmetry
of the graph of sin x to find the following.
0

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