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Lesson 30: The Definite Integral

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Matthew Leingang
Matthew Leingang

We define the definite integral as a limit of Riemann sums, compute some approximations, then investigate the basic additive and comparative properties

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Section 5.2 The Definite Integral Math 1a December 7, 2007 Announcements my next office hours: Monday 1–2, Tuesday 3–4 (SC 323) MT II is graded. You’ll get it back today Final seview sessions: Wed 1/9 and Thu 1/10 in Hall D, Sun 1/13 in Hall C, all 7–8:30pm Final tentatively scheduled for January 17
Outline The definite integral as a limit Estimating the Definite Integral Properties of the integral Comparison 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 to b is the number n b f (x) dx = lim f (ci ) ∆x ∆x→0 a i=1
Notation/Terminology b f (x) dx a
Notation/Terminology b f (x) dx a — integral sign (swoopy S)
Notation/Terminology b f (x) dx a — integral sign (swoopy S) f (x) — integrand
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)
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?)
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
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
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 ∆x = and xi = a + i ∆x n
Outline The definite integral as a limit Estimating the Definite Integral Properties of the integral Comparison Properties of the Integral
Estimating the Definite Integral Given a partition of [a, b] into n pieces, let xi be the midpoint of ¯ [xi−1 , xi ]. Define n Mn = f (¯i ) ∆x. x i=1
Example 1 4 Estimate dx using the midpoint rule and four divisions. 1 + x2 0
Example 1 4 Estimate dx using the midpoint rule and four divisions. 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)
Example 1 4 Estimate dx using the midpoint rule and four divisions. 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) 1 4 4 4 4 = + + + 4 65/64 73/64 89/64 113/64
Example 1 4 Estimate dx using the midpoint rule and four divisions. 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) 1 4 4 4 4 = + + + 4 65/64 73/64 89/64 113/64 150, 166, 784 ≈ 3.1468 = 47, 720, 465
Outline The definite integral as a limit Estimating the Definite Integral Properties of the integral Comparison Properties of the Integral
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
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 2. [f (x) + g (x)] dx = f (x) dx + g (x) dx. a a a
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 2. [f (x) + g (x)] dx = f (x) dx + g (x) dx. a a a b b 3. cf (x) dx = c f (x) dx. a a
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 2. [f (x) + g (x)] dx = f (x) dx + g (x) dx. a a a b b 3. cf (x) dx = c f (x) dx. a a b b b [f (x) − g (x)] dx = f (x) dx − 4. g (x) dx. a a a
More Properties of the Integral Conventions: a b f (x) dx = − f (x) dx b a
More Properties of the Integral Conventions: a b f (x) dx = − f (x) dx b a a f (x) dx = 0 a
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 5. f (x) dx = f (x) dx + f (x) dx for all a, b, and c. a a b
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 (b) f (x) dx. 4
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
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
Outline The definite integral as a limit Estimating the Definite Integral Properties of the integral Comparison 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 all x in [a, b], then b f (x) dx ≥ 0 a
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
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
Example 2 1 Estimate dx using the comparison properties. x 1
Example 2 1 Estimate dx using the comparison properties. x 1 Solution Since 1 1 ≤x ≤ 2 1 for all x in [1, 2], we have 2 1 1 ·1≤ dx ≤ 1 · 1 2 x 1

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Lesson 30: The Definite Integral

  • 1. Section 5.2 The Definite Integral Math 1a December 7, 2007 Announcements my next office hours: Monday 1–2, Tuesday 3–4 (SC 323) MT II is graded. You’ll get it back today Final seview sessions: Wed 1/9 and Thu 1/10 in Hall D, Sun 1/13 in Hall C, all 7–8:30pm Final tentatively scheduled for January 17
  • 2. Outline The definite integral as a limit Estimating the Definite Integral Properties of the integral Comparison Properties of the Integral
  • 3. 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 n b f (x) dx = lim f (ci ) ∆x ∆x→0 a i=1
  • 4. Notation/Terminology b f (x) dx a
  • 5. Notation/Terminology b f (x) dx a — integral sign (swoopy S)
  • 6. Notation/Terminology b f (x) dx a — integral sign (swoopy S) f (x) — integrand
  • 7. 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)
  • 8. 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?)
  • 9. 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
  • 10. 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
  • 11. 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 ∆x = and xi = a + i ∆x n
  • 12. Outline The definite integral as a limit Estimating the Definite Integral Properties of the integral Comparison Properties of the Integral
  • 13. Estimating the Definite Integral Given a partition of [a, b] into n pieces, let xi be the midpoint of ¯ [xi−1 , xi ]. Define n Mn = f (¯i ) ∆x. x i=1
  • 14. Example 1 4 Estimate dx using the midpoint rule and four divisions. 1 + x2 0
  • 15. Example 1 4 Estimate dx using the midpoint rule and four divisions. 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)
  • 16. Example 1 4 Estimate dx using the midpoint rule and four divisions. 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) 1 4 4 4 4 = + + + 4 65/64 73/64 89/64 113/64
  • 17. Example 1 4 Estimate dx using the midpoint rule and four divisions. 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) 1 4 4 4 4 = + + + 4 65/64 73/64 89/64 113/64 150, 166, 784 ≈ 3.1468 = 47, 720, 465
  • 18. Outline The definite integral as a limit Estimating the Definite Integral Properties of the integral Comparison Properties of the Integral
  • 19. 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
  • 20. 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 2. [f (x) + g (x)] dx = f (x) dx + g (x) dx. a a a
  • 21. 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 2. [f (x) + g (x)] dx = f (x) dx + g (x) dx. a a a b b 3. cf (x) dx = c f (x) dx. a a
  • 22. 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 2. [f (x) + g (x)] dx = f (x) dx + g (x) dx. a a a b b 3. cf (x) dx = c f (x) dx. a a b b b [f (x) − g (x)] dx = f (x) dx − 4. g (x) dx. a a a
  • 23. More Properties of the Integral Conventions: a b f (x) dx = − f (x) dx b a
  • 24. More Properties of the Integral Conventions: a b f (x) dx = − f (x) dx b a a f (x) dx = 0 a
  • 25. 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 5. f (x) dx = f (x) dx + f (x) dx for all a, b, and c. a a b
  • 26. 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 (b) f (x) dx. 4
  • 27. 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
  • 28. 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
  • 29. Outline The definite integral as a limit Estimating the Definite Integral Properties of the integral Comparison Properties of the Integral
  • 30. Comparison Properties of the Integral Theorem Let f and g be integrable functions on [a, b].
  • 31. 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
  • 32. 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
  • 33. 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
  • 34. Example 2 1 Estimate dx using the comparison properties. x 1
  • 35. Example 2 1 Estimate dx using the comparison properties. x 1 Solution Since 1 1 ≤x ≤ 2 1 for all x in [1, 2], we have 2 1 1 ·1≤ dx ≤ 1 · 1 2 x 1