Numerical Approximations of Integrals

1
Lesson 7-7

• Numerical Approximations of Integrals
• -- What we do when we cant integrate a function
• Riemann Sums
• Trapezoidal Rule

2
Strategies for Integrals that we cant do

• Use Riemann Sums left-hand or right-hand, to
  approximate the area under the curve (i.e., the
  value of the definite integral)
• Use the Trapezoidal Rule to approximate the area
  under the curve
• The trapezoidal rule gives us an average of the
  right and left hand Riemann Sum approximations
• The graph of f The trapezoidal rule will
  concave down underestimate the area concave up
  overestimate the area.

3
Trapezoidal Rule
From Geometry Area of a Trapezoid ½h(b1
b2) Bases will the height or the functional
value and h is the width, (b-a)/n, of each
trapezoid Since only the first base and the last
base will be used only once ? Area ½h(b1
2b2 2bn-1 bn)
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Example 1
Use the trapezoidal rule with n 4 to approximate
h (b-a)/n (p-0)/4 p/4 n
f(x) 0 sin(0) 0 1
sin(p/4) 0.7071 2 sin(p/2) 1 3
sin(3p/4) 0.7071 4 sin(p) 0
Area ½ h (b1 b2)
Area1 ½ h (b1 b2) p/8(0 0.7071) Area2 ½
h (b1 b2) p/8(0.7071 1) Area3 ½ h (b1
b2) p/8(1 0.7071) Area4 ½ h (b1 b2)
p/8(0.7071 0) AreaT ? Ai
(p/8)2(0.7071) 2(1.7071)
1.89612
2
3
1
4
5
Example 2
Use the trapezoidal rule with n 5 to approximate
h (b-a)/n (1-0)/5 1/5 n
f(x) 0 f(0) 1 1 f(1/5)
1.2214 2 f(2/5) 1.4918 3
f(3/5) 1.8221 4 f(4/5) 2.2255 5
f(1) 2.7183
Area ½ h (b1 b2)
Area1 ½ h (b1 b2) 1/10(1 1.2214) Area2
½ h (b1 b2) 1/101.2214 1.4918) Area3 ½ h
(b1 b2) 1/10(1.4918 1.8221) Area4 ½ h (b1
b2) 1/10(1.8221 2.2255) Area5 ½ h (b1
b2) 1/10(2.2255 2.7183) AreaT ? Ai
(1/10)1 2(1.2214) 2(1.4918)
2(1.8221) 2(2.2255) 2.7183
1.72399
1
note calculator did a numeric apx
6
Pond Example
h 20 (b-a)/n n 7
Area ½ h (b1 b2)
Area1 ½ h (b1 b2) 10(0 40) 400 Area2
½ h (b1 b2) 10(40 50) 900 Area3 ½ h
(b1 b2) 10(50 45) 950 Area4 ½ h (b1
b2) 10(45 30) 750 Area5 ½ h (b1 b2)
10(30 45) 750 Area6 ½ h (b1 b2) 10(45
40) 850 Area7 ½ h (b1 b2) 10(40
0) 400 AreaT ? Ai

5000
7
Approximating the Error of the Estimate

• If f has a continuous second derivative on a,b,
  then the error E in approximating
  by the trapezoidal rule is

8
Example 3
Use the error formula to find the maximum
possible error in approximating the integral,
with n 4.
f(x) (x 1)-1 f(x) -(x1)-2 f(x)
2/(x1)-3
(b-a)³ E -------- max f (x)
where a x b 12n²
(1-0)³ E -------- 2 (at x 0
) 124² E 0.01042
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Example 4
Use the error formula to find n so that the error
in the approximation of the definite integral is
less than 0.00001.
f(x) (x 1)-1 f(x) -(x1)-2 f(x)
2/(x1)-3
(b a)³ n² ---------- max f (x)
(always round up to 12 E
next whole when
calculating n)
(1 0)³ 2 n² -------------------
(always round up to 12
(0.00001) next whole when

calculating n) n² 16666.67 n 129.1 ? n
130
10
Summary Homework

• Summary
• Riemann Sums can be used to get approximations to
  definite integrals that we dont know how to
  integrate
• The error used to approximate the integral can be
  calculated
• The number of sub-intervals required to keep the
  error bounded can be calculated
• Homework
• pg 527 529 7, 8, 9