1. Packet #23 The Definite Integral
Math 180
Packet #23
The Definite Integral

2. Sigma notation helps us write sums compactly
Sigma notation helps us write sums compactly. Here are some examples: = 𝑖=1 6 𝑖 2 𝑓 1 +𝑓 2 +𝑓 3 +…+𝑓 100 = 𝑖=1 100 𝑓(𝑖) 𝑖=1 𝑛 𝑎 𝑖 = 𝑎 1 + 𝑎 2 + 𝑎 3 +…+ 𝑎 𝑛−1 + 𝑎 𝑛

3. Sigma notation helps us write sums compactly
Sigma notation helps us write sums compactly. Here are some examples: = 𝑖=1 6 𝑖 2 𝑓 1 +𝑓 2 +𝑓 3 +…+𝑓 100 = 𝑖=1 100 𝑓(𝑖) 𝑖=1 𝑛 𝑎 𝑖 = 𝑎 1 + 𝑎 2 + 𝑎 3 +…+ 𝑎 𝑛−1 + 𝑎 𝑛

4. Sigma notation helps us write sums compactly
Sigma notation helps us write sums compactly. Here are some examples: = 𝑖=1 6 𝑖 2 𝑓 1 +𝑓 2 +𝑓 3 +…+𝑓 100 = 𝑖=1 100 𝑓(𝑖) 𝑖=1 𝑛 𝑎 𝑖 = 𝑎 1 + 𝑎 2 + 𝑎 3 +…+ 𝑎 𝑛−1 + 𝑎 𝑛

5. Sigma notation helps us write sums compactly
Sigma notation helps us write sums compactly. Here are some examples: = 𝑖=1 6 𝑖 2 𝑓 1 +𝑓 2 +𝑓 3 +…+𝑓 100 = 𝑖=1 100 𝑓(𝑖) 𝑖=1 𝑛 𝑎 𝑖 = 𝑎 1 + 𝑎 2 + 𝑎 3 +…+ 𝑎 𝑛−1 + 𝑎 𝑛

6. Sigma notation helps us write sums compactly
Sigma notation helps us write sums compactly. Here are some examples: = 𝑖=1 6 𝑖 2 𝑓 1 +𝑓 2 +𝑓 3 +…+𝑓 100 = 𝑖=1 100 𝑓(𝑖) 𝑖=1 𝑛 𝑎 𝑖 = 𝑎 1 + 𝑎 2 + 𝑎 3 +…+ 𝑎 𝑛−1 + 𝑎 𝑛

7. Sigma notation helps us write sums compactly
Sigma notation helps us write sums compactly. Here are some examples: = 𝑖=1 6 𝑖 2 𝑓 1 +𝑓 2 +𝑓 3 +…+𝑓 100 = 𝑖=1 100 𝑓(𝑖) 𝑖=1 𝑛 𝑎 𝑖 = 𝑎 1 + 𝑎 2 + 𝑎 3 +…+ 𝑎 𝑛−1 + 𝑎 𝑛

8. 𝑖=1 3 −1 𝑖 cos 𝑖𝜋 = −1 1 cos 1⋅𝜋 + −1 2 cos 2⋅𝜋 + −1 3 cos 3⋅𝜋 =1+1+1 =3

9. 𝑖=1 3 −1 𝑖 cos 𝑖𝜋 = −1 1 cos 1⋅𝜋 + −1 2 cos 2⋅𝜋 + −1 3 cos 3⋅𝜋 =1+1+1 =3

10. 𝑖=1 3 −1 𝑖 cos 𝑖𝜋 = −1 1 cos 1⋅𝜋 + −1 2 cos 2⋅𝜋 + −1 3 cos 3⋅𝜋 =1+1+1 =3

11. 𝑖=1 3 −1 𝑖 cos 𝑖𝜋 = −1 1 cos 1⋅𝜋 + −1 2 cos 2⋅𝜋 + −1 3 cos 3⋅𝜋 =1+1+1 =3

12. Note: Since we can rearrange terms, we have that 𝑖=1 3 𝑖+ 𝑖 2 = = = 𝑖=1 3 𝑖 + 𝑖=1 3 𝑖 2

13. Note: Since we can rearrange terms, we have that 𝑖=1 3 𝑖+ 𝑖 2 = = = 𝑖=1 3 𝑖 + 𝑖=1 3 𝑖 2

14. Note: Since we can rearrange terms, we have that 𝑖=1 3 𝑖+ 𝑖 2 = = = 𝑖=1 3 𝑖 + 𝑖=1 3 𝑖 2

15. Note: Since we can rearrange terms, we have that 𝑖=1 3 𝑖+ 𝑖 2 = = = 𝑖=1 3 𝑖 + 𝑖=1 3 𝑖 2

16. In general, 𝒊=𝟏 𝒏 𝒂 𝒊 ± 𝒃 𝒊 = 𝒊=𝟏 𝒏 𝒂 𝒊 ± 𝒊=𝟏 𝒏 𝒃 𝒊 𝒊=𝟏 𝒏 𝒄 𝒂 𝒊 =𝒄⋅ 𝒊=𝟏 𝒏 𝒂 𝒊 (for example, 𝑖= 𝑖 =2 𝑖=1 10 𝑖 ) 𝒊=𝟏 𝒏 𝒄 =𝒏⋅𝒄 (for example, 𝑖= =99⋅2=198)

17. In general, 𝒊=𝟏 𝒏 𝒂 𝒊 ± 𝒃 𝒊 = 𝒊=𝟏 𝒏 𝒂 𝒊 ± 𝒊=𝟏 𝒏 𝒃 𝒊 𝒊=𝟏 𝒏 𝒄 𝒂 𝒊 =𝒄⋅ 𝒊=𝟏 𝒏 𝒂 𝒊 (for example, 𝑖= 𝑖 =2 𝑖=1 10 𝑖 ) 𝒊=𝟏 𝒏 𝒄 =𝒏⋅𝒄 (for example, 𝑖= =99⋅2=198)

18. In general, 𝒊=𝟏 𝒏 𝒂 𝒊 ± 𝒃 𝒊 = 𝒊=𝟏 𝒏 𝒂 𝒊 ± 𝒊=𝟏 𝒏 𝒃 𝒊 𝒊=𝟏 𝒏 𝒄 𝒂 𝒊 =𝒄⋅ 𝒊=𝟏 𝒏 𝒂 𝒊 (for example, 𝑖= 𝑖 =2 𝑖=1 10 𝑖 ) 𝒊=𝟏 𝒏 𝒄 =𝒏⋅𝒄 (for example, 𝑖= =99⋅2=198)

19. In general, 𝒊=𝟏 𝒏 𝒂 𝒊 ± 𝒃 𝒊 = 𝒊=𝟏 𝒏 𝒂 𝒊 ± 𝒊=𝟏 𝒏 𝒃 𝒊 𝒊=𝟏 𝒏 𝒄 𝒂 𝒊 =𝒄⋅ 𝒊=𝟏 𝒏 𝒂 𝒊 (for example, 𝑖= 𝑖 =2 𝑖=1 10 𝑖 ) 𝒊=𝟏 𝒏 𝒄 =𝒏⋅𝒄 (for example, 𝑖= =99⋅2=198)

20. It will be useful to know closed forms for the following summations: 𝒊=𝟏 𝒏 𝒊 = 𝒏 𝒏+𝟏 𝟐 𝒊=𝟏 𝒏 𝒊 𝟐 = 𝒏 𝒏+𝟏 (𝟐𝒏+𝟏) 𝟔 𝒊=𝟏 𝒏 𝒊 𝟑 = 𝒏 𝒏+𝟏 𝟐 𝟐

21. Ex 1. Evaluate the following sums. 𝑖=1 15 𝑖 3 𝑖=1 10 𝑖 2𝑖+1

22. Ex 2. Write a formula for the estimation of the area under the curve 𝑓 𝑥 = 𝑥 2 from 𝑥=0 to 𝑥=1 using 𝑛 rectangles and right endpoints. Then take the limit as 𝑛→∞.

25. Riemann Sums
Summations like 1 𝑛 2 1 𝑛 + 2 𝑛 2 1 𝑛 +…+ 𝑛 𝑛 2 1 𝑛 = 𝑖=1 𝑛 𝑘 𝑛 2 ⋅ 1 𝑛 are called Riemann Sums. If the rectangles have the same widths, then the general Riemann Sum can be written: 𝑖=1 𝑛 𝑓 𝑥 𝑖 ∗ Δ𝑥 Here, 𝑥 𝑖 ∗ is any 𝑥-value inside each subinterval 𝑥 𝑖−1 , 𝑥 𝑖 . (For right endpoints, 𝑥 𝑖 ∗ =𝑎+𝑖Δ𝑥.) So, 𝑓( 𝑥 𝑖 ∗ ) is the height of each rectangle. Δ𝑥 is the width of each rectangle (calculated by 𝑏−𝑎 𝑛 ).

28. If the rectangles have different widths, then the even more general Riemann Sum is: 𝑖=1 𝑛 𝑓 𝑥 𝑖 ∗ Δ 𝑥 𝑖 The difference here is that the rectangle widths, Δ 𝑥 𝑖 , are indexed by 𝑖, and so can be different.

30. Notes:
No matter what each rectangle width is, as long as the maximum rectangle width (called the norm) approaches 0 as 𝑛→∞, then lim 𝑛→∞ 𝑖=1 𝑛 𝑓 𝑥 𝑖 ∗ Δ 𝑥 𝑖 =Net area
Rectangles below the 𝑥-axis count as the negative of the area in the sum. We often say that we’re estimating “the area under the curve 𝑦=𝑓(𝑥)”, but we’re actually estimating the net area, where area above the 𝑥-axis is counted positive, and area below the 𝑥-axis is counted negative.

31. Notes:
No matter what each rectangle width is, as long as the maximum rectangle width (called the norm) approaches 0 as 𝑛→∞, then lim 𝑛→∞ 𝑖=1 𝑛 𝑓 𝑥 𝑖 ∗ Δ 𝑥 𝑖 =Net area
Rectangles below the 𝑥-axis count as the negative of the area in the sum. We often say that we’re estimating “the area under the curve 𝑦=𝑓(𝑥)”, but we’re actually estimating the net area, where area above the 𝑥-axis is counted positive, and area below the 𝑥-axis is counted negative.

32. We use the integral symbol to represent the infinite sum of infinitely thin rectangles: 𝑎 𝑏 𝑓(𝑥) 𝑑𝑥= lim 𝑛→∞ 𝑖=1 𝑛 𝑓 𝑥 𝑖 ∗ Δ𝑥 𝑎 𝑏 𝑓(𝑥) 𝑑𝑥 is called a _______________, and is read: ”the integral from 𝑎 to 𝑏 of 𝑓 of 𝑥 dee 𝑥” 𝑎 is the _______________________. 𝑏 is the _______________________. 𝑓(𝑥) is called the ___________.

33. We use the integral symbol to represent the infinite sum of infinitely thin rectangles: 𝑎 𝑏 𝑓(𝑥) 𝑑𝑥= lim 𝑛→∞ 𝑖=1 𝑛 𝑓 𝑥 𝑖 ∗ Δ𝑥 𝑎 𝑏 𝑓(𝑥) 𝑑𝑥 is called a _______________, and is read: ”the integral from 𝑎 to 𝑏 of 𝑓 of 𝑥 dee 𝑥” 𝑎 is the _______________________. 𝑏 is the _______________________. 𝑓(𝑥) is called the ___________.
definite integral

34. lower limit of integration
We use the integral symbol to represent the infinite sum of infinitely thin rectangles: 𝑎 𝑏 𝑓(𝑥) 𝑑𝑥= lim 𝑛→∞ 𝑖=1 𝑛 𝑓 𝑥 𝑖 ∗ Δ𝑥 𝑎 𝑏 𝑓(𝑥) 𝑑𝑥 is called a _______________, and is read: ”the integral from 𝑎 to 𝑏 of 𝑓 of 𝑥 dee 𝑥” 𝑎 is the _______________________. 𝑏 is the _______________________. 𝑓(𝑥) is called the ___________.
definite integral
lower limit of integration

35. lower limit of integration upper limit of integration
We use the integral symbol to represent the infinite sum of infinitely thin rectangles: 𝑎 𝑏 𝑓(𝑥) 𝑑𝑥= lim 𝑛→∞ 𝑖=1 𝑛 𝑓 𝑥 𝑖 ∗ Δ𝑥 𝑎 𝑏 𝑓(𝑥) 𝑑𝑥 is called a _______________, and is read: ”the integral from 𝑎 to 𝑏 of 𝑓 of 𝑥 dee 𝑥” 𝑎 is the _______________________. 𝑏 is the _______________________. 𝑓(𝑥) is called the ___________.
definite integral
lower limit of integration
upper limit of integration

36. lower limit of integration upper limit of integration integrand
We use the integral symbol to represent the infinite sum of infinitely thin rectangles: 𝑎 𝑏 𝑓(𝑥) 𝑑𝑥= lim 𝑛→∞ 𝑖=1 𝑛 𝑓 𝑥 𝑖 ∗ Δ𝑥 𝑎 𝑏 𝑓(𝑥) 𝑑𝑥 is called a _______________, and is read: ”the integral from 𝑎 to 𝑏 of 𝑓 of 𝑥 dee 𝑥” 𝑎 is the _______________________. 𝑏 is the _______________________. 𝑓(𝑥) is called the ___________.
definite integral
lower limit of integration
upper limit of integration
integrand

37. Note: 𝑎 𝑏 𝑓(𝑥) 𝑑𝑥= 𝑎 𝑏 𝑓 𝑡 𝑑𝑡= 𝑎 𝑏 𝑓(𝑢) 𝑑𝑢

38. Note: 𝑎 𝑏 𝑓(𝑥) 𝑑𝑥= 𝑎 𝑏 𝑓 𝑡 𝑑𝑡= 𝑎 𝑏 𝑓(𝑢) 𝑑𝑢

39. Theorem: If 𝑓 is continuous over 𝑎,𝑏 (with possibly a finite number of jump discontinuities), then 𝑎 𝑏 𝑓(𝑥) 𝑑𝑥 exists and so 𝑓 is said to be integrable over 𝑎,𝑏 .

40. Properties: 1. 𝑏 𝑎 𝑓(𝑥) 𝑑𝑥=− 𝑎 𝑏 𝑓(𝑥) 𝑑𝑥 2. 𝑎 𝑎 𝑓(𝑥) 𝑑𝑥= ____
1. 𝑏 𝑎 𝑓(𝑥) 𝑑𝑥=− 𝑎 𝑏 𝑓(𝑥) 𝑑𝑥
2. 𝑎 𝑎 𝑓(𝑥) 𝑑𝑥= ____
3. 𝑎 𝑏 𝑘𝑓 𝑥 𝑑𝑥=𝑘 𝑎 𝑏 𝑓 𝑥 𝑑𝑥
4. 𝑎 𝑏 𝑓 𝑥 ±𝑔 𝑥 𝑑𝑥= 𝑎 𝑏 𝑓 𝑥 𝑑𝑥± 𝑎 𝑏 𝑔 𝑥 𝑑𝑥

41. Properties: 1. 𝑏 𝑎 𝑓(𝑥) 𝑑𝑥=− 𝑎 𝑏 𝑓(𝑥) 𝑑𝑥 2. 𝑎 𝑎 𝑓(𝑥) 𝑑𝑥= ____
1. 𝑏 𝑎 𝑓(𝑥) 𝑑𝑥=− 𝑎 𝑏 𝑓(𝑥) 𝑑𝑥
2. 𝑎 𝑎 𝑓(𝑥) 𝑑𝑥= ____
3. 𝑎 𝑏 𝑘𝑓 𝑥 𝑑𝑥=𝑘 𝑎 𝑏 𝑓 𝑥 𝑑𝑥
4. 𝑎 𝑏 𝑓 𝑥 ±𝑔 𝑥 𝑑𝑥= 𝑎 𝑏 𝑓 𝑥 𝑑𝑥± 𝑎 𝑏 𝑔 𝑥 𝑑𝑥

42. Properties: 1. 𝑏 𝑎 𝑓(𝑥) 𝑑𝑥=− 𝑎 𝑏 𝑓(𝑥) 𝑑𝑥 2. 𝑎 𝑎 𝑓(𝑥) 𝑑𝑥= ____
1. 𝑏 𝑎 𝑓(𝑥) 𝑑𝑥=− 𝑎 𝑏 𝑓(𝑥) 𝑑𝑥
2. 𝑎 𝑎 𝑓(𝑥) 𝑑𝑥= ____
3. 𝑎 𝑏 𝑘𝑓 𝑥 𝑑𝑥=𝑘 𝑎 𝑏 𝑓 𝑥 𝑑𝑥
4. 𝑎 𝑏 𝑓 𝑥 ±𝑔 𝑥 𝑑𝑥= 𝑎 𝑏 𝑓 𝑥 𝑑𝑥± 𝑎 𝑏 𝑔 𝑥 𝑑𝑥 𝟎

43. Properties: 1. 𝑏 𝑎 𝑓(𝑥) 𝑑𝑥=− 𝑎 𝑏 𝑓(𝑥) 𝑑𝑥 2. 𝑎 𝑎 𝑓(𝑥) 𝑑𝑥= ____
1. 𝑏 𝑎 𝑓(𝑥) 𝑑𝑥=− 𝑎 𝑏 𝑓(𝑥) 𝑑𝑥
2. 𝑎 𝑎 𝑓(𝑥) 𝑑𝑥= ____
3. 𝑎 𝑏 𝑘𝑓 𝑥 𝑑𝑥=𝑘 𝑎 𝑏 𝑓 𝑥 𝑑𝑥
4. 𝑎 𝑏 𝑓 𝑥 ±𝑔 𝑥 𝑑𝑥= 𝑎 𝑏 𝑓 𝑥 𝑑𝑥± 𝑎 𝑏 𝑔 𝑥 𝑑𝑥 𝟎

44. Properties: 5. 𝑎 𝑏 𝑓(𝑥) 𝑑𝑥+ 𝑏 𝑐 𝑓(𝑥) 𝑑𝑥= 𝑎 𝑐 𝑓(𝑥) 𝑑𝑥 6
Properties: 5. 𝑎 𝑏 𝑓(𝑥) 𝑑𝑥+ 𝑏 𝑐 𝑓(𝑥) 𝑑𝑥= 𝑎 𝑐 𝑓(𝑥) 𝑑𝑥 6. min 𝑓 ⋅ 𝑏−𝑎 ≤ 𝑎 𝑏 𝑓 𝑥 𝑑𝑥≤ max 𝑓 ⋅ 𝑏−𝑎 7. If 𝑓 𝑥 ≥𝑔(𝑥) on 𝑎,𝑏 , then 𝑎 𝑏 𝑓(𝑥) 𝑑𝑥≥ 𝑎 𝑏 𝑔(𝑥) 𝑑𝑥.

45. Properties: 5. 𝑎 𝑏 𝑓(𝑥) 𝑑𝑥+ 𝑏 𝑐 𝑓(𝑥) 𝑑𝑥= 𝑎 𝑐 𝑓(𝑥) 𝑑𝑥 6
Properties: 5. 𝑎 𝑏 𝑓(𝑥) 𝑑𝑥+ 𝑏 𝑐 𝑓(𝑥) 𝑑𝑥= 𝑎 𝑐 𝑓(𝑥) 𝑑𝑥 6. min 𝑓 ⋅ 𝑏−𝑎 ≤ 𝑎 𝑏 𝑓 𝑥 𝑑𝑥≤ max 𝑓 ⋅ 𝑏−𝑎 7. If 𝑓 𝑥 ≥𝑔(𝑥) on 𝑎,𝑏 , then 𝑎 𝑏 𝑓(𝑥) 𝑑𝑥≥ 𝑎 𝑏 𝑔(𝑥) 𝑑𝑥.

46. Properties: 5. 𝑎 𝑏 𝑓(𝑥) 𝑑𝑥+ 𝑏 𝑐 𝑓(𝑥) 𝑑𝑥= 𝑎 𝑐 𝑓(𝑥) 𝑑𝑥 6
Properties: 5. 𝑎 𝑏 𝑓(𝑥) 𝑑𝑥+ 𝑏 𝑐 𝑓(𝑥) 𝑑𝑥= 𝑎 𝑐 𝑓(𝑥) 𝑑𝑥 6. min 𝑓 ⋅ 𝑏−𝑎 ≤ 𝑎 𝑏 𝑓 𝑥 𝑑𝑥≤ max 𝑓 ⋅ 𝑏−𝑎 7. If 𝑓 𝑥 ≥𝑔(𝑥) on 𝑎,𝑏 , then 𝑎 𝑏 𝑓(𝑥) 𝑑𝑥≥ 𝑎 𝑏 𝑔(𝑥) 𝑑𝑥.

47. Properties: 5. 𝑎 𝑏 𝑓(𝑥) 𝑑𝑥+ 𝑏 𝑐 𝑓(𝑥) 𝑑𝑥= 𝑎 𝑐 𝑓(𝑥) 𝑑𝑥 6
Properties: 5. 𝑎 𝑏 𝑓(𝑥) 𝑑𝑥+ 𝑏 𝑐 𝑓(𝑥) 𝑑𝑥= 𝑎 𝑐 𝑓(𝑥) 𝑑𝑥 6. min 𝑓 ⋅ 𝑏−𝑎 ≤ 𝑎 𝑏 𝑓 𝑥 𝑑𝑥≤ max 𝑓 ⋅ 𝑏−𝑎 7. If 𝑓 𝑥 ≥𝑔(𝑥) on 𝑎,𝑏 , then 𝑎 𝑏 𝑓(𝑥) 𝑑𝑥≥ 𝑎 𝑏 𝑔(𝑥) 𝑑𝑥.

48. Ex 3. Suppose that −1 1 𝑓 𝑥 𝑑𝑥=5, 1 4 𝑓(𝑥) 𝑑𝑥=−2, and −1 1 ℎ(𝑥) 𝑑𝑥=7
Ex 3. Suppose that −1 1 𝑓 𝑥 𝑑𝑥=5, 1 4 𝑓(𝑥) 𝑑𝑥=−2, and −1 1 ℎ(𝑥) 𝑑𝑥=7. Find the following. 4 1 𝑓 𝑥 𝑑𝑥 −1 1 2𝑓 𝑥 +3ℎ 𝑥 𝑑𝑥 −1 4 𝑓 𝑥 𝑑𝑥

49. Definite Integrals as Areas
For a nonnegative, integrable function 𝑓(𝑥), 𝑎 𝑏 𝑓(𝑥) 𝑑𝑥 calculates the area under the graph of 𝑓. If 𝑓(𝑥) ever negative in 𝑎, 𝑏 , then 𝑎 𝑏 𝑓 𝑥 𝑑𝑥 computes net area.

50. Definite Integrals as Areas
For a nonnegative, integrable function 𝑓(𝑥), 𝑎 𝑏 𝑓(𝑥) 𝑑𝑥 calculates the area under the graph of 𝑓. If 𝑓(𝑥) is ever negative in 𝑎, 𝑏 , then 𝑎 𝑏 𝑓 𝑥 𝑑𝑥 computes net area.

51. Ex 4. Graph the integrand and use the area under the graph to evaluate the integral. −4 0 16− 𝑥 2 𝑑𝑥

52. Ex 4. Graph the integrand and use the area under the graph to evaluate the integral. −4 0 16− 𝑥 2 𝑑𝑥

53. Ex 4. Graph the integrand and use the area under the graph to evaluate the integral. −4 0 16− 𝑥 2 𝑑𝑥

54. Ex 4. Graph the integrand and use the area under the graph to evaluate the integral. −4 0 16− 𝑥 2 𝑑𝑥

55. Ex 4. Graph the integrand and use the area under the graph to evaluate the integral. −4 0 16− 𝑥 2 𝑑𝑥

56. Ex 4. Graph the integrand and use the area under the graph to evaluate the integral. −4 0 16− 𝑥 2 𝑑𝑥

57. Ex 5. Graph the integrand and use the area under the graph to evaluate the integral. −4 0 (1+ 16− 𝑥 2 ) 𝑑𝑥

58. Ex 5. Graph the integrand and use the area under the graph to evaluate the integral. −4 0 (1+ 16− 𝑥 2 ) 𝑑𝑥

59. Ex 5. Graph the integrand and use the area under the graph to evaluate the integral. −4 0 (1+ 16− 𝑥 2 ) 𝑑𝑥

60. Ex 5. Graph the integrand and use the area under the graph to evaluate the integral. −4 0 (1+ 16− 𝑥 2 ) 𝑑𝑥

61. Ex 5. Graph the integrand and use the area under the graph to evaluate the integral. −4 0 (1+ 16− 𝑥 2 ) 𝑑𝑥