Section	5.2
          The	Definite	Integral

                   V63.0121, Calculus	I



                      April	15, 200...
Outline


  Recall


  The	definite	integral	as	a	limit


  Estimating	the	Definite	Integral


  Properties	of	the	integral
...
Cavalieri’s	method	in	general
   Let f be	a	positive	function	defined	on	the	interval [a, b]. We	want
   to	find	the	area	be...
Forming	Riemann	sums
  We	have	many	choices	of	representative	points	to	approximate
  the	area	in	each	subinterval.



 le...
Forming	Riemann	sums
  We	have	many	choices	of	representative	points	to	approximate
  the	area	in	each	subinterval.



  r...
Forming	Riemann	sums
  We	have	many	choices	of	representative	points	to	approximate
  the	area	in	each	subinterval.



  m...
Forming	Riemann	sums
  We	have	many	choices	of	representative	points	to	approximate
  the	area	in	each	subinterval.




  ...
Theorem	of	the	(previous)	Day



 Theorem
 If f is	a	continuous	function	on
 [a, b] or	has	finitely	many	jump
 discontinuit...
Theorem	of	the	(previous)	Day



 Theorem
 If f is	a	continuous	function	on
 [a, b] or	has	finitely	many	jump
 discontinuit...
Theorem	of	the	(previous)	Day



 Theorem
 If f is	a	continuous	function	on
 [a, b] or	has	finitely	many	jump
 discontinuit...
Theorem	of	the	(previous)	Day



 Theorem
 If f is	a	continuous	function	on
 [a, b] or	has	finitely	many	jump
 discontinuit...
Theorem	of	the	(previous)	Day



 Theorem
 If f is	a	continuous	function	on
 [a, b] or	has	finitely	many	jump
 discontinuit...
Theorem	of	the	(previous)	Day



 Theorem
 If f is	a	continuous	function	on
 [a, b] or	has	finitely	many	jump
 discontinuit...
Theorem	of	the	(previous)	Day



 Theorem
 If f is	a	continuous	function	on
 [a, b] or	has	finitely	many	jump
 discontinuit...
Theorem	of	the	(previous)	Day



 Theorem
 If f is	a	continuous	function	on
 [a, b] or	has	finitely	many	jump
 discontinuit...
Theorem	of	the	(previous)	Day



 Theorem
 If f is	a	continuous	function	on
 [a, b] or	has	finitely	many	jump
 discontinuit...
Theorem	of	the	(previous)	Day



 Theorem
 If f is	a	continuous	function	on
 [a, b] or	has	finitely	many	jump
 discontinuit...
Theorem	of	the	(previous)	Day



 Theorem
 If f is	a	continuous	function	on
 [a, b] or	has	finitely	many	jump
 discontinuit...
Theorem	of	the	(previous)	Day



 Theorem
 If f is	a	continuous	function	on
 [a, b] or	has	finitely	many	jump
 discontinuit...
Theorem	of	the	(previous)	Day



 Theorem
 If f is	a	continuous	function	on
 [a, b] or	has	finitely	many	jump
 discontinuit...
Theorem	of	the	(previous)	Day



 Theorem
 If f is	a	continuous	function	on
 [a, b] or	has	finitely	many	jump
 discontinuit...
Theorem	of	the	(previous)	Day



 Theorem
 If f is	a	continuous	function	on
 [a, b] or	has	finitely	many	jump
 discontinuit...
Theorem	of	the	(previous)	Day



 Theorem
 If f is	a	continuous	function	on
 [a, b] or	has	finitely	many	jump
 discontinuit...
Theorem	of	the	(previous)	Day



 Theorem
 If f is	a	continuous	function	on
 [a, b] or	has	finitely	many	jump
 discontinuit...
Theorem	of	the	(previous)	Day



 Theorem
 If f is	a	continuous	function	on
 [a, b] or	has	finitely	many	jump
 discontinuit...
Theorem	of	the	(previous)	Day



 Theorem
 If f is	a	continuous	function	on
 [a, b] or	has	finitely	many	jump
 discontinuit...
Theorem	of	the	(previous)	Day



 Theorem
 If f is	a	continuous	function	on
 [a, b] or	has	finitely	many	jump
 discontinuit...
Theorem	of	the	(previous)	Day



 Theorem
 If f is	a	continuous	function	on
 [a, b] or	has	finitely	many	jump
 discontinuit...
Theorem	of	the	(previous)	Day



 Theorem
 If f is	a	continuous	function	on
 [a, b] or	has	finitely	many	jump
 discontinuit...
Theorem	of	the	(previous)	Day



 Theorem
 If f is	a	continuous	function	on
 [a, b] or	has	finitely	many	jump
 discontinuit...
Theorem	of	the	(previous)	Day



 Theorem
 If f is	a	continuous	function	on
 [a, b] or	has	finitely	many	jump
 discontinuit...
Theorem	of	the	(previous)	Day



 Theorem
 If f is	a	continuous	function	on
 [a, b] or	has	finitely	many	jump
 discontinuit...
Theorem	of	the	(previous)	Day



 Theorem
 If f is	a	continuous	function	on
 [a, b] or	has	finitely	many	jump
 discontinuit...
Theorem	of	the	(previous)	Day



 Theorem
 If f is	a	continuous	function	on
 [a, b] or	has	finitely	many	jump
 discontinuit...
Theorem	of	the	(previous)	Day



 Theorem
 If f is	a	continuous	function	on
 [a, b] or	has	finitely	many	jump
 discontinuit...
Theorem	of	the	(previous)	Day



 Theorem
 If f is	a	continuous	function	on
 [a, b] or	has	finitely	many	jump
 discontinuit...
Theorem	of	the	(previous)	Day



 Theorem
 If f is	a	continuous	function	on
 [a, b] or	has	finitely	many	jump
 discontinuit...
Outline


  Recall


  The	definite	integral	as	a	limit


  Estimating	the	Definite	Integral


  Properties	of	the	integral
...
The	definite	integral	as	a	limit




   Definition
   If f is	a	function	defined	on [a, b], the definite	integral	of f from a
...
Notation/Terminology


                       ∫   b
                               f(x) dx
                       a




  ...
Notation/Terminology


                             ∫    b
                                      f(x) dx
                 ...
Notation/Terminology


                             ∫    b
                                      f(x) dx
                 ...
Notation/Terminology


                             ∫     b
                                       f(x) dx
               ...
Notation/Terminology


                             ∫     b
                                       f(x) dx
               ...
Notation/Terminology


                             ∫     b
                                       f(x) dx
               ...
The	limit	can	be	simplified

   Theorem
   If f is	continuous	on [a, b] or	if f has	only	finitely	many	jump
   discontinuiti...
The	limit	can	be	simplified

   Theorem
   If f is	continuous	on [a, b] or	if f has	only	finitely	many	jump
   discontinuiti...
Outline


  Recall


  The	definite	integral	as	a	limit


  Estimating	the	Definite	Integral


  Properties	of	the	integral
...
Estimating	the	Definite	Integral




   Given	a	partition	of [a, b] into n pieces, let ¯i be	the	midpoint	of
              ...
Example
           ∫   1
                     4
                          dx using	the	midpoint	rule	and	four	divisions.
E...
Example
           ∫       1
                         4
                              dx using	the	midpoint	rule	and	four	...
Example
           ∫   1
                     4
                          dx using	the	midpoint	rule	and	four	divisions.
E...
Example
           ∫   1
                     4
                          dx using	the	midpoint	rule	and	four	divisions.
E...
Outline


  Recall


  The	definite	integral	as	a	limit


  Estimating	the	Definite	Integral


  Properties	of	the	integral
...
Properties	of	the	integral


   Theorem	(Additive	Properties	of	the	Integral)
   Let f and g be	integrable	functions	on [a...
Properties	of	the	integral


   Theorem	(Additive	Properties	of	the	Integral)
   Let f and g be	integrable	functions	on [a...
Properties	of	the	integral


   Theorem	(Additive	Properties	of	the	Integral)
   Let f and g be	integrable	functions	on [a...
Properties	of	the	integral


   Theorem	(Additive	Properties	of	the	Integral)
   Let f and g be	integrable	functions	on [a...
More	Properties	of	the	Integral



   Conventions:   ∫                      ∫
                       a                    ...
More	Properties	of	the	Integral



   Conventions:   ∫                         ∫
                       a                 ...
More	Properties	of	the	Integral



   Conventions:           ∫                             ∫
                             ...
Example
Suppose f and g are	functions	with
    ∫4
       f(x) dx = 4
     0
    ∫5
       f(x) dx = 7
     0
    ∫5
      ...
Solution
We	have
(a)
           ∫                               ∫                   ∫
               5                    ...
Solution
We	have
(a)
           ∫                                       ∫                         ∫
               5      ...
Outline


  Recall


  The	definite	integral	as	a	limit


  Estimating	the	Definite	Integral


  Properties	of	the	integral
...
Comparison	Properties	of	the	Integral
   Theorem
   Let f and g be	integrable	functions	on [a, b].




                   ...
Comparison	Properties	of	the	Integral
   Theorem
   Let f and g be	integrable	functions	on [a, b].
    6. If f(x) ≥ 0 for	...
Comparison	Properties	of	the	Integral
   Theorem
   Let f and g be	integrable	functions	on [a, b].
    6. If f(x) ≥ 0 for	...
Comparison	Properties	of	the	Integral
   Theorem
   Let f and g be	integrable	functions	on [a, b].
    6. If f(x) ≥ 0 for	...
Example
           ∫   2
                   1
                     dx using	the	comparison	properties.
Estimate
          ...
Example
           ∫   2
                   1
                     dx using	the	comparison	properties.
Estimate
          ...
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Lesson 24: The Definite Integral (Section 10 version)

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The limit of Riemann Sums has a name: the definite integral. We compute a few "easy" ones and show general properties.

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Transcript of "Lesson 24: The Definite Integral (Section 10 version)"

  1. 1. Section 5.2 The Definite Integral V63.0121, Calculus I April 15, 2009 Announcements My office is now WWH 624 Final Exam Friday, May 8, 2:00–3:50pm . . . . . .
  2. 2. Outline Recall The definite integral as a limit Estimating the Definite Integral Properties of the integral Comparison Properties of the Integral . . . . . .
  3. 3. Cavalieri’s method in general Let f be a positive function defined on the interval [a, b]. We want to find the area between x = a, x = b, y = 0, and y = f(x). For each positive integer n, divide up the interval into n pieces. b−a . For each i between 1 and n, let xi be the ith Then ∆x = n step between a and b. So x0 = a b−a x1 = x0 + ∆x = a + n b−a x2 = x1 + ∆x = a + 2 · ... n b−a xi = a + i · ... n b−a xn = a + n · =b n .. . .. x . . 0 . 1 . . . . i . . .. n−1. n x x. x. x x . . . . . .
  4. 4. Forming Riemann sums We have many choices of representative points to approximate the area in each subinterval. left endpoints… n ∑ Ln = f(xi−1 )∆x i=1 ....... x . In general, choose ci to be a point in the ith interval [xi−1 , xi ]. Form the Riemann sum n ∑ Sn = f(c1 )∆x + f(c2 )∆x + · · · + f(cn )∆x = f(ci )∆x i=1 . . . . . .
  5. 5. Forming Riemann sums We have many choices of representative points to approximate the area in each subinterval. right endpoints… n ∑ Rn = f(xi )∆x i=1 ....... x . In general, choose ci to be a point in the ith interval [xi−1 , xi ]. Form the Riemann sum n ∑ Sn = f(c1 )∆x + f(c2 )∆x + · · · + f(cn )∆x = f(ci )∆x i=1 . . . . . .
  6. 6. Forming Riemann sums We have many choices of representative points to approximate the area in each subinterval. midpoints… ∑ ( xi−1 + xi ) n Mn = f ∆x 2 i=1 ....... x . In general, choose ci to be a point in the ith interval [xi−1 , xi ]. Form the Riemann sum n ∑ Sn = f(c1 )∆x + f(c2 )∆x + · · · + f(cn )∆x = f(ci )∆x i=1 . . . . . .
  7. 7. Forming Riemann sums We have many choices of representative points to approximate the area in each subinterval. random points… ....... x . In general, choose ci to be a point in the ith interval [xi−1 , xi ]. Form the Riemann sum n ∑ Sn = f(c1 )∆x + f(c2 )∆x + · · · + f(cn )∆x = f(ci )∆x i=1 . . . . . .
  8. 8. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has finitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we made. . . . . . .
  9. 9. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has finitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we made. . . . . . .
  10. 10. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has finitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . . . x . matter what choice of ci we made. . . . . . .
  11. 11. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has finitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . . . . x . matter what choice of ci we made. . . . . . .
  12. 12. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has finitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . . . . . x . matter what choice of ci we made. . . . . . .
  13. 13. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has finitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . . . . . . x . matter what choice of ci we made. . . . . . .
  14. 14. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has finitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no ....... x . matter what choice of ci we made. . . . . . .
  15. 15. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has finitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no ........ x . matter what choice of ci we made. . . . . . .
  16. 16. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has finitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no ......... x . matter what choice of ci we made. . . . . . .
  17. 17. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has finitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no .......... x . matter what choice of ci we made. . . . . . .
  18. 18. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has finitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no ........... x . matter what choice of ci we made. . . . . . .
  19. 19. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has finitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no ............ x . matter what choice of ci we made. . . . . . .
  20. 20. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has finitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no ............. x . matter what choice of ci we made. . . . . . .
  21. 21. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has finitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no .............. x . matter what choice of ci we made. . . . . . .
  22. 22. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has finitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no ............... x . matter what choice of ci we made. . . . . . .
  23. 23. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has finitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no ................ x . matter what choice of ci we made. . . . . . .
  24. 24. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has finitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no ................. x . matter what choice of ci we made. . . . . . .
  25. 25. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has finitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no .................. x . matter what choice of ci we made. . . . . . .
  26. 26. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has finitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no ................... x . matter what choice of ci we made. . . . . . .
  27. 27. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has finitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no .................... x . matter what choice of ci we made. . . . . . .
  28. 28. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has finitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no ..................... . x . matter what choice of ci we made. . . . . . .
  29. 29. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has finitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no ...................... . x . matter what choice of ci we made. . . . . . .
  30. 30. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has finitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no ....................... . x . matter what choice of ci we made. . . . . . .
  31. 31. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has finitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no ........................ . x . matter what choice of ci we made. . . . . . .
  32. 32. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has finitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no ......................... . x . matter what choice of ci we made. . . . . . .
  33. 33. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has finitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no .......................... . x . matter what choice of ci we made. . . . . . .
  34. 34. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has finitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no ........................... . x . matter what choice of ci we made. . . . . . .
  35. 35. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has finitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no ............................ . x . matter what choice of ci we made. . . . . . .
  36. 36. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has finitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no ............................. . x . matter what choice of ci we made. . . . . . .
  37. 37. Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] or has finitely many jump discontinuities, then {n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no .............................. . x . matter what choice of ci we made. . . . . . .
  38. 38. Outline Recall The definite integral as a limit Estimating the Definite Integral Properties of the integral Comparison Properties of the Integral . . . . . .
  39. 39. The definite integral as a limit Definition If f is a function defined on [a, b], the definite integral of f from a to b is the number ∫b n ∑ f(x) dx = lim f(ci ) ∆x ∆x→0 a i=1 . . . . . .
  40. 40. Notation/Terminology ∫ b f(x) dx a . . . . . .
  41. 41. Notation/Terminology ∫ b f(x) dx a ∫ — integral sign (swoopy S) . . . . . .
  42. 42. Notation/Terminology ∫ b f(x) dx a ∫ — integral sign (swoopy S) f(x) — integrand . . . . . .
  43. 43. Notation/Terminology ∫ b f(x) dx a ∫ — integral sign (swoopy S) f(x) — integrand a and b — limits of integration (a is the lower limit and b the upper limit) . . . . . .
  44. 44. Notation/Terminology ∫ b f(x) dx a ∫ — integral sign (swoopy S) f(x) — integrand a and b — limits of integration (a is the lower limit and b the upper limit) dx — ??? (a parenthesis? an infinitesimal? a variable?) . . . . . .
  45. 45. Notation/Terminology ∫ b f(x) dx a ∫ — integral sign (swoopy S) f(x) — integrand a and b — limits of integration (a is the lower limit and b the upper limit) dx — ??? (a parenthesis? an infinitesimal? a variable?) The process of computing an integral is called integration or quadrature . . . . . .
  46. 46. The limit can be simplified Theorem If f is continuous on [a, b] or if f has only finitely many jump discontinuities, then f is integrable on [a, b]; that is, the definite ∫b integral f(x) dx exists. a . . . . . .
  47. 47. The limit can be simplified Theorem If f is continuous on [a, b] or if f has only finitely many jump discontinuities, then f is integrable on [a, b]; that is, the definite ∫b integral f(x) dx exists. a Theorem If f is integrable on [a, b] then ∫ n ∑ b f(x) dx = lim f(xi )∆x, n→∞ a i=1 where b−a and xi = a + i ∆x ∆x = n . . . . . .
  48. 48. Outline Recall The definite integral as a limit Estimating the Definite Integral Properties of the integral Comparison Properties of the Integral . . . . . .
  49. 49. Estimating the Definite Integral Given a partition of [a, b] into n pieces, let ¯i be the midpoint of x [xi−1 , xi ]. Define n ∑ Mn = f(¯i ) ∆x. x i=1 . . . . . .
  50. 50. Example ∫ 1 4 dx using the midpoint rule and four divisions. Estimate 1 + x2 0 . . . . . .
  51. 51. Example ∫ 1 4 dx using the midpoint rule and four divisions. Estimate 1 + x2 0 Solution 1 1 3 The partition is 0 < < < < 1, so the estimate is 4 2 4 ( ) 1 4 4 4 4 M4 = + + + 2 2 2 1 + (7/8)2 4 1 + (1/8) 1 + (3/8) 1 + (5/8) . . . . . .
  52. 52. Example ∫ 1 4 dx using the midpoint rule and four divisions. Estimate 1 + x2 0 Solution 1 1 3 The partition is 0 < < < < 1, so the estimate is 4 2 4 ( ) 1 4 4 4 4 M4 = + + + 4 1 + (1/8)2 1 + (3/8)2 1 + (5/8)2 1 + (7/8)2 ( ) 1 4 4 4 4 = + + + 4 65/64 73/64 89/64 113/64 . . . . . .
  53. 53. Example ∫ 1 4 dx using the midpoint rule and four divisions. Estimate 1 + x2 0 Solution 1 1 3 The partition is 0 < < < < 1, so the estimate is 4 2 4 ( ) 1 4 4 4 4 M4 = + + + 4 1 + (1/8)2 1 + (3/8)2 1 + (5/8)2 1 + (7/8)2 ( ) 1 4 4 4 4 = + + + 4 65/64 73/64 89/64 113/64 150, 166, 784 ≈ 3.1468 = 47, 720, 465 . . . . . .
  54. 54. Outline Recall The definite integral as a limit Estimating the Definite Integral Properties of the integral Comparison Properties of the Integral . . . . . .
  55. 55. Properties of the integral Theorem (Additive Properties of the Integral) Let f and g be integrable functions on [a, b] and c a constant. Then ∫b c dx = c(b − a) 1. a . . . . . .
  56. 56. Properties of the integral Theorem (Additive Properties of the Integral) Let f and g be integrable functions on [a, b] and c a constant. Then ∫b c dx = c(b − a) 1. a ∫ ∫ ∫ b b b [f(x) + g(x)] dx = f(x) dx + g(x) dx. 2. a a a . . . . . .
  57. 57. Properties of the integral Theorem (Additive Properties of the Integral) Let f and g be integrable functions on [a, b] and c a constant. Then ∫b c dx = c(b − a) 1. a ∫ ∫ ∫ b b b [f(x) + g(x)] dx = f(x) dx + g(x) dx. 2. a a a ∫ ∫ b b cf(x) dx = c f(x) dx. 3. a a . . . . . .
  58. 58. Properties of the integral Theorem (Additive Properties of the Integral) Let f and g be integrable functions on [a, b] and c a constant. Then ∫b c dx = c(b − a) 1. a ∫ ∫ ∫ b b b [f(x) + g(x)] dx = f(x) dx + g(x) dx. 2. a a a ∫ ∫ b b cf(x) dx = c f(x) dx. 3. a a ∫ ∫ ∫ b b b [f(x) − g(x)] dx = f(x) dx − g(x) dx. 4. a a a . . . . . .
  59. 59. More Properties of the Integral Conventions: ∫ ∫ a b f(x) dx = − f(x) dx b a . . . . . .
  60. 60. More Properties of the Integral Conventions: ∫ ∫ a b f(x) dx = − f(x) dx b a ∫ a f(x) dx = 0 a . . . . . .
  61. 61. More Properties of the Integral Conventions: ∫ ∫ a b f(x) dx = − f(x) dx b a ∫ a f(x) dx = 0 a This allows us to have ∫c ∫b ∫ c f(x) dx = f(x) dx + f(x) dx for all a, b, and c. 5. a a b . . . . . .
  62. 62. Example Suppose f and g are functions with ∫4 f(x) dx = 4 0 ∫5 f(x) dx = 7 0 ∫5 g(x) dx = 3. 0 Find ∫5 [2f(x) − g(x)] dx (a) 0 ∫5 f(x) dx. (b) 4 . . . . . .
  63. 63. Solution We have (a) ∫ ∫ ∫ 5 5 5 [2f(x) − g(x)] dx = 2 f(x) dx − g(x) dx 0 0 0 = 2 · 7 − 3 = 11 . . . . . .
  64. 64. Solution We have (a) ∫ ∫ ∫ 5 5 5 [2f(x) − g(x)] dx = 2 f(x) dx − g(x) dx 0 0 0 = 2 · 7 − 3 = 11 (b) ∫ ∫ ∫ 5 5 4 f(x) dx − f(x) dx = f(x) dx 4 0 0 =7−4=3 . . . . . .
  65. 65. Outline Recall The definite integral as a limit Estimating the Definite Integral Properties of the integral Comparison Properties of the Integral . . . . . .
  66. 66. Comparison Properties of the Integral Theorem Let f and g be integrable functions on [a, b]. . . . . . .
  67. 67. Comparison Properties of the Integral Theorem Let f and g be integrable functions on [a, b]. 6. If f(x) ≥ 0 for all x in [a, b], then ∫ b f(x) dx ≥ 0 a . . . . . .
  68. 68. Comparison Properties of the Integral Theorem Let f and g be integrable functions on [a, b]. 6. If f(x) ≥ 0 for all x in [a, b], then ∫ b f(x) dx ≥ 0 a 7. If f(x) ≥ g(x) for all x in [a, b], then ∫ ∫ b b f(x) dx ≥ g(x) dx a a . . . . . .
  69. 69. Comparison Properties of the Integral Theorem Let f and g be integrable functions on [a, b]. 6. If f(x) ≥ 0 for all x in [a, b], then ∫ b f(x) dx ≥ 0 a 7. If f(x) ≥ g(x) for all x in [a, b], then ∫ ∫ b b f(x) dx ≥ g(x) dx a a 8. If m ≤ f(x) ≤ M for all x in [a, b], then ∫ b m(b − a) ≤ f(x) dx ≤ M(b − a) a . . . . . .
  70. 70. Example ∫ 2 1 dx using the comparison properties. Estimate x 1 . . . . . .
  71. 71. Example ∫ 2 1 dx using the comparison properties. Estimate x 1 Solution Since 1 1 ≤x≤ 2 1 for all x in [1, 2], we have ∫ 2 1 1 ·1≤ dx ≤ 1 · 1 x 2 1 . . . . . .
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