2. Definite Integral and Numeric Integration
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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
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Remember sigma notation from pre-calculus?
The sum of n terms a1 , a2 , a3 ,..., an is written as S 
The i tells you where to start and end summing.
n
a
i 1
i
Approximating Area
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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.
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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
• 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)
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The more rectangles you make, the better the approximation
o
If a function is increasing, LRAM will underestimate the area and RRAM will overestimate it.
o
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.
Example 2
Find the area under y=x2 from x=0 to x=3, use width of ½. Use all methods
Example 3
Find the area under y=x2+2x-3 from x=0 to x=2, use width of ½
How many rectangles should we make?
The estimate of area gets more and more accurate as the number of rectangles (n) gets larger
4
 2xdx
0
If we take the limit as n approaches infinity, we should get the exact area
We will talk more about this tomorrow…..
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
ia
a
A  lim  f ( xi )x   f ( x )dx
n 
Example 4
4
Evaluate
 2xdx
geometrically as well as on your calculator
0
Negative Area?
• The example we just looked at was non-negative 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
b
definite integral of f from a to b
 f ( x)dx
a
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If f(x) is negative and integrable over a closed interval [a,b], then the area under the curve is the
b
OPPOSITE of the definite integral of f from a to b.
  f ( x ) dx
a
b
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In general,
 f ( x)dx
does NOT give us area but rather the NET accumulation over the interval x=a to x=b.
a
b
If f(x) is positive and negative on a closed interval, then
 f ( x)dx
will NOT give us area.
a
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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.
When integrating left to right, regions above the x-axis are positive and regions below the x-axis are
negative.
b
c
b
a
a
c
 f ( x)dx   f ( x)dx   f ( x)dx
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When integrating right to left, regions above the x-axis are negative and regions below the x-axis are
positive.
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This can be summarized as
b
a
a
b
 f ( x)dx   f ( x)dx
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
b
The property that allows us to do the calculations in the previous example is

a
c
b
a
c
f ( x)dx   f ( x )dx   f ( x )dx
5
Example 6
( x 2  4)dx
Approximate
using four subintervals of equal length and trapezoidal method. Test your answer
1
against the calculator’s approximation using fnint. Can any of these approximations represent the area of the
region? Why or why not?

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   f ( x) dx
a
Example 8
We can also find areas when our function is given to us in either data form or graph form.
3
Approximate
f’(1)
 f ( x)dx
using LRAM, MRAM, RRAM, and trapezoids. Do these represent area? Also approximate
0
8
Example 9
f ( x)dx
Approximate 1
using LRAM, RRAM, and trapezoids. Do these represent area? Why did I leave off MRAM?
Also approximate f’(7)

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
 f ( x)dx
Evaluate 7
and
the area of the region?
5
.
 f ( x)dx
Do these represent the area of the region? Why or why not? If not, what is
2
Properties of Definite Integrals
We have seen some of these already
Example 12
Given that
1

4
f ( x)dx  5,
1

1
1
Find
 f ( x)dx
4
4
 f ( x)dx
1
1
 2 f ( x)  3h( x)dx
1
1
f ( x)dx  2,
 h( x)dx  7
1
1
 f ( x)dx
0
2
 h( x)dx
2
Example 13
10
If
8
 f ( x)dx  17
and
0
 f ( x)dx  12
0
, find
10
 (3 f ( x)  2)dx
8
Example 14
 9  x2 
Find  
dx
x 3 
5 
6
using area of the region to help evaluate the integral.
Example 15

If
 sin xdx  2
0
use this fact and symmetry of the graph of sin x to find the following.