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The Definite Integral as Area - Classwork

Instead of using the expression "the area under the curve f ( x) between x = a and x = b, we will now denote a shorthand to represent the same thing. We will use what is called "a definite integral." The definite integral sign is the same as the indefinite integral sign ( ! ) but will contain two limits of integration. The form is as follows:

b

, f ( x) dx . While this does not seems to make much sense, there is a reason for it. The f ( x) represents the height

a

of any one rectangle while the dx represents the width of any one rectangle. So f ( x) dx means the area of any one rectangle. The integral represents the sum of an infinite number of these rectangles. The a represents the starting place for these rectangles while the b represents the ending place for these integrals.

f (x )

a

dx f ( x ) dx

b

a

b

, f ( x ) dx

a

b

The area of one rectangle = f ( x) dx

The sum of an infinite number of rectangle areas. Each rectangle is infinitely thin.

When a < b, we are determining the area under the curve from left to right. In that case, our dx is a positive

b

number. If f ( x) is above the axis, then

, f ( x) dx will be a positive number.

a b

When b < a, we are determining the area under the curve from right to left. In that case, our dx is a negative number. If f ( x) is above the axis, then

, f ( x) dx will be a negative number. This can be summarized below:

a

f ( x) > 0 (curve above axis)

f ( x) < 0 (curve below axis)

b

dx > 0 (left to right) (a < b) dx < 0 (right to left) (b < a)

b

,

a b a

f ( x) dx > 0 (Area positive)

(Area negative)

, f ( x) dx < 0

a b

(Area negative) (Area positive)

, f ( x) dx < 0

, f ( x) dx > 0

a

Furthermore, there are three more rules which will make sense to you:

a

1. 2. 3.

, f ( x) dx

a b a

- If we start at a and end at a, there is no area.

,

a b

a

f ( x) dx = "

c

, f ( x) dx

b

c a

- From a to b gives an area. From b to a gives the negative of this area. - Total the area from a to b, add area from b to c = the area from a to c.

- 143 Stu Schwartz

, f ( x ) dx + , f ( x ) dx = , f ( x ) dx

b

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Example) Below you are given the graph of f ( x) formed by lines and a semi-circle. Find the definite integrals.

This is the graph of

f ( x)

4

1

3

1. 4. 7. 10.

, f (t ) dt = 0

4 3

2. 5. 8.

, f (t ) dt = 2

0 6 3 10

3. 6. 9. 12. 15. 18. 21.

8

, f (t ) dt = 2

1 3

, f (t ) dt = 4

0 6

, f (t ) dt = "1.5 , f (t ) dt = 26 0 10 "3

, f (t ) dt = 1.5

6 6 10

, f (t ) dt = 2.5

0 10

, f (t ) dt = "20 "1 "3

, f (t ) dt = 2.5 + 20 0 "3 0

11. " 14. 17. 20.

5 5

, f (t ) dt = "2.5 " 2, f (t ) dt = "11 , f (t ) dt = 13 + 210 "4

, f (t ) dt = 3 , f (t ) dt = ".5

, f (t ) dt = 2.5 + 20

13. " 16. 19.

, f (t ) dt = 11 , f (t ) dt = 10.5 , f (t ) dt = 5.5 + 20 5 "2

0 10

"4 10

"4 10

"4

"4

, f (t ) dt = 13 + 2h( x) dx = "11, and

10

,

2 f ( t ) dt = "34 " 4 -

Suppose

5

,

f ( x ) dx = 18 ,

"2

,

g ( x) dx = 5,

"2 5

,

"2

, f ( x) dx = 0, find

24.

"2 5

22.

"2 5

, ( f ( x ) + g ( x )) dx = 23

, (g( x ) + 2) dx = 19

3

23.

"2 5

, [ f ( x ) + g ( x ) " h( x )] dx = 34

, ( f ( x ) " 6) dx = "24

8

, 4 g( x ) dx = "20 , h( x " 2) dx = "11

0 7

25.

26.

27.

"2

"2

28.

"4

, g( x + 2) dx = 5

29.

, f ( x ) dx = "18

5

30.

, [ f ( x " 3) + 3] dx = 39

1

8

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- 144 -

Stu Schwartz

The Definite Integral as Area - Homework

f ( x)

1

4

4

1. 4. 7. 10. 13. 16. 19. 22.

,

0 5

f ( x ) dx = 2 f ( x ) dx = 0 f ( x ) dx = 0

0

2. 5. 8. 11. 14. 17. 20. 23.

2

,

2 5

f ( x ) dx = 8 f ( x ) dx = 2 f ( x ) dx = 14

0

3. 6. 9. 12. 15. 18. 21. 24.

,

1 6

f ( x ) dx = 12 f ( x ) dx = "2 f ( x ) dx = "4

0

,

5 6

,

4 6

,

5 2

,

4

,

0

,

3

5 "3

, , ,

6

f ( x ) dx = "16 f ( x ) dx = 6 f ( x ) dx =

94

,

6 2 "3 "6

f ( x ) dx = "14 f ( x ) dx = 0 f ( x ) dx =

"94

0 "3

, , ,

6 1

"3 "3

, ,

4 0

f ( x ) dx = "6 f ( x ) dx = "8 f ( x ) dx =

9"6 4

"6

"3 "6

"6 1

,

"6 1

, ,

9f ( x ) dx = +8 4

9f ( x ) dx = " " 8 4 - f ( x ) dx = 6

4

"2

,

f ( x ) dx = 2

f ( x ) dx = 6

2

"2

"2

,

"6

, f ( x ) dx = 94- + 24

1

6

Suppose that

4

, f ( x) dx = 2, , f ( x) dx = "1, , f ( x) dx = 7,

0 1 2 4

evaluate the following: 27.

25.

,

1 1

f ( x ) dx = 6 f ( x + 1) dx = "1

3 3

26.

,

0 2 0

3 f ( x ) dx = 27

, f ( x ) dx = 3

0 4

28.

,

0

29.

, ( f ( x ) + 3) dx = 8

30.

, f ( x " 2) dx = 2

2

31. If

, f ( x) dx = "1, find , f ( x) dx

0 "3

if f is a) even -2 b) odd 0

- 145 Stu Schwartz

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The problems on this page and the next should be done without calculators. 32. For each function below, use a sketch of f to evaluate the 4 definite integrals on top given the statement on the left. (a is a positive constant)

2 4 1

, f ( x) dx

0

, f ( x) dx

1

, f ( x) dx

5

"9

, f ( x ) dx

9

f ( x) = 3 f ( x) = a f ( x) = " a

6 2a "2 a

9 3a "3 a

--12 "4 a 4 a

54 18 a 18 a

33. Evaluate the following by making a sketch of the function. 3 3 "1 25 1 25 a. ( x + 2) dx = + = 12 b. x + 2 dx = + = 13 2 2 2 2 "3 "3

,

,

3

c.

"3

,

. 91 x + 2 dx = 2 6 + = 21 0 23 / 2

3

d.

"3

,

. 11 2 " x dx = 2 2 + = 5 0 23 / 2

34. Evaluate

1 + x, , f ( x ) dx where f ( x ) = 62 " x , 5

0

2

7

if 0 4 x 4 1 = 2 . Make a sketch of the function. if 1 < x 4 2

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Stu Schwartz

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