1

Warm-Up

a) Write the standard equation of a circle centered at the origin with a radius of 2.

b) Write the equation for the top half of the circle.

c) Write the equation for the bottom half of the circle.

Integration4

Copyright © Cengage Learning. All rights reserved.

Riemann Sums and Definite Integrals

2015

Copyright © Cengage Learning. All rights reserved.

4.3

4

Understand the definition of a Riemann sum.

Evaluate a definite integral using limits.

Evaluate a definite integral using properties of definite integrals.

Objectives

5

Riemann Sums

9

Riemann Sums

The region shown in Figure 4.18 has an area of . Find the area of the region bounded by the graph of

Figure 4.18

2 , 0, 0.x y y x

10

Because the square bounded by 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1 has an area of 1, you can conclude that the area of the region shown in Figure 4.17 has an area of .

Figure 4.17

Riemann Sums

13

Riemann SumsThe sum of the areas of rectangles on an interval between a curve and an axis is called a Riemann sum. We use these to approximate the area under a curve, on an interval.

By taking the limit of a Riemann sum as the number of rectangles goes to infinity, we can find the actual area. This is true because as n increases, the width of each or the largest subinterval approaches zero.

This is a key feature of the development of definite integrals.

1

lim [ , ] n

ini

f c x on a b

Sum of areas of very skinny rectangles under the curve

17

Definite Integrals

19

Definite Integrals

20

Definition of a definite integral:

Sum of areas of very skinny rectangles under the curve

1

lim [ , ] ( ) bn

ini a

f c x on a b f x dx

21

Definite Integrals

22

Example 2 – Evaluating a Definite Integral as a Limit

Use the limit definition to evaluate the definite integral

Solution:The function f(x) = 2x is integrable on the interval [–2, 1] because it is continuous on [–2, 1]. Work follows.

23

Example 2 – Solution

For computational convenience, define x by subdividing [–2, 1] into n subintervals of equal width

Choosing ci as the right endpoint of each subinterval produces

cont’d

24

Example 2 – Solution

So, the definite integral is given by

cont’d

25

Because the definite integral in Example 2 is negative, it does not represent the area of the region shown in Figure 4.20.

Definite integrals can be positive, negative, or zero.

For a definite integral to be interpreted as an area, the function f must be continuous and nonnegative on [a, b].

Could we have used a Geometric area formula? Figure 4.20

Definite Integrals

26

Definite Integrals

29

In the last lesson we evaluated definite integrals by using the limit definition of a Riemann sum. As a short-cut, we can check to see whether the definite integral represents the area of a common geometric region such as a rectangle, triangle, or semicircle.

Definite Integrals

30

Example 3 – Areas of Common Geometric Figures

Sketch the region corresponding to each definite integral. Then evaluate each integral using a geometric formula.

a.

b.

c.

31

Example 3(a) – Solution

This region is a rectangle of height 4 and width 2.

Figure 4.23(a)

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Example 3(b) – Solution

This region is a trapezoid with an altitude of 3 and parallel bases of lengths 2 and 5. The formula for the area of a trapezoid is h(b1 + b2). (on its side b(h1 + h2)/2.)

Figure 4.23(b)

cont’d

33

Example 3(c) – Solution

This region is a semicircle of radius 2. The formula for the area of a semicircle is

Figure 4.23(c)

cont’d

34

Properties of Definite Integrals

36

Properties of Definite Integrals

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Example 4 – Evaluating Definite Integrals

a. Because the sine function is defined at x = π, and the upper and lower limits of integration are equal, you can write

b. The integral has a value of you can write: 3

0

2x dx

38

cont’dExample 4 – Evaluating Definite Integrals

39

Example 5 – Using the Additive Interval Property

40

Properties of Definite Integrals

Note that Property 2 of Theorem 4.7 can be extended to cover any finite number of functions. For example,

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Example 6 – Evaluation of a Definite Integral

Evaluate using each of the given values.

Solution:

44

1

0

)a f x dx 4

3

) 3b f x dx 11

0

)c f x dx

The graph of f is shown above. Evaluate each definite integral:

45

Example 9

8

0

f x dx

4, 4, 4x

f xx x

The function f is defined below. Use geometric formulas to find:

46

Example 8 – Evaluation of a Definite Integral

Sketch the region whose area is given by:

Use a geometric formula to evaluate the integral.

32

0

9 x dx

49

AB Homework Section 4.3 pg.278 #7, 13-49 odd, 47,49

Day 2: MMM 145-146

50

BC Homework Section 4.3 pg.278 #23-49 odd + MMM 145