11 warm-up: (let h be measured in feet) h(t) = -5t 2 + 20t + 15 a. estimate the instantaneous...

11

Warm-Up: (let h be measured in feet) h(t) = -5t2 + 20t + 15

a. Estimate the instantaneous velocity of the ball 3 seconds after it’s thrown.

b. Estimate the acceleration of the ball 3 seconds after it’s thrown.

c. Estimate the maximum height.

v(t) = -10t + 20

v(3) = -30 + 20 = -10 ft/s

a(t) = -10 ft/ s2

0 = -10t + 20

t = 2

h(2) = 35 feet

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Area and the Fundamental Theorem of Calculus

5.4

Day 1 Homework: p. 348/9: 2-8 EVEN, 18-28 EVENDay 2 Homework: p. 348: 10-16 EVEN, p. 349: 48-56 EVEN

33

Understand the relationship between area and definite integrals.

Evaluate definite integrals using the Fundamental Theorem of Calculus.

Use definite integrals to solve marginal analysis problems.

Find the average values of functions over closed intervals.

Use properties of even and odd functions to help evaluate definite integrals.

Objectives

44

Rules for Definite Integrals

Order of Integration

Zero

Constant Multiple

b

a

a

b

dxxfdxxf )()(

0)( a

a

dxxf

b

a

b

a

dxxfkdxxkf )()(

b

a

b

a

dxxfdxxf )()(

55

Example 1 Using the Rules for Definite Integrals

Suppose

Find each of the following integrals, if possible.

.14 and,11,93

4

5

3

3

4

dxxhdxxfdxxf

3

5

dxxf 5

3

dxxf

1111

66


Suppose


.14 and,11,93

4

5

3

3

4

dxxhdxxfdxxf

5

4

dxxf

5

3

3

4

dxxfdxxf

119 2

77


Suppose


.14 and,11,93

4

5

3

3

4

dxxhdxxfdxxf

4

3

dxxf

Not possible; not enough information given.

88

From your study of geometry, you know that area is a number that defines the size of a bounded region. For simple regions, such as rectangles, triangles, and circles, area can be found using geometric formulas.

In this section, you will learn how to use calculus to find the areas of nonstandard regions, such as the region R shown in Figure 5.5.

Area and Definite Integrals

Figure 5.5

99

Example 2: Use the graph of the integrand and areas to evaluate the integral

.)32/(4

2

dxx

2 5

6A=1/2(6)(2+5)=

21

1010

Example 2 – Evaluating a Definite Integral Using a Geometric Formula

The definite integral

represents the area of the region bounded by the graph of f (x) = 2x, the x-axis, and theline x = 2, as shown in Figure 5.6.

Figure 5.6

1111

Example 3 – Evaluating a Definite Integral Using a Geometric Formula

The region is triangular, with a height of 4 units and a base of 2 units. Using the formula for the area of a triangle, you have

cont’d

1212

If f is any function that is continuous on a closed interval [a, b] then the definite integral of f (x) from a to b is defined to be

where F is an antiderivative of f. Remember that definite integrals do not necessarily represent areas and can be negative, zero, or positive.

The Fundamental Theorem of Calculus

1313

Example 4: Evaluating a Definite Integral

dx

u = x – 2

du = dx

du

-

dx

• No substitution needed here.

= 0

=

0 – = -

u = 3 - 2

u = 0 - 2

1414

Closure: Area Under a Curve

Determine the area under the curve over the interval [–4, 4].

216 xy

2

2

1rA

242

1 A

8A

1515

Warm-Up: v(t) = 800 – 32t. Find the distance traveled between 8 and 20 seconds.

Solution: Velocity is the derivative of distance, so you can graph the velocity function and find the area under the line to the x-axis.

The shape is a trapezoid.

A = ½ h (b1 + b1 )

h = 20 -8 = 12

b1 = v(8) =544

b2 = v(20) = 160

A = ½ (12)(544 + 160)

= 4224 units

1616

Warm-Up, p. 349: 32

Let u =

du = 4x dx;

u(2) = and u(0) = = 1

= du = =

= 1.5 and = .5

1.5 - .5 = 1

1717

Area and Definite Integrals

1818

Example 1 – Finding Area by the Fundamental Theorem

Find the area of the region bounded by the x-axis and the graph of

Solution:

Note that on the interval as shown in Figure 5.9.

Figure 5.9

1919

Example 1 – Solution

So, you can represent the area of the region by a definite integral. To find the area, use the Fundamental Theorem of Calculus.

cont’d

2020


So, the area of the region is square units.

cont’d

2121

Example 2 – Interpreting Absolute Value

Evaluate

Solution:

The region represented by the definite integral is shown in Figure 5.11.

Figure 5.11

2222


Using Property 3 of definite integrals, rewrite the integral as two definite integrals.

cont’d

2323

In this section, you will examine the reverse process. That is, you will be given the marginal cost, marginal revenue, or marginal profit and you will use a definite integral to find the exact increase or decrease in cost, revenue, or profit obtained by selling one or several additional units.

Marginal Analysis

2424

Example 3 – Analyzing a Profit Function

The marginal profit for a product is modeled by

a. Find the change in profit when sales increase from 100 to 101 units.

b. Find the change in profit when sales increase from 100 to 110 units.

2525

Example 3(a) – Solution

The change in profit obtained by increasing sales from 100 to 101 units is

2626

Example 3(b) – Solution

The change in profit obtained by increasing sales from 100 to 110 units is

2727

Average (Mean) Value

If f is integrable on the interval [a, b], the function’s average (mean) value on the interval is

.1

b

a

dxxfab

fav

2828

Example 4 – Finding the Average Cost

The cost per unit c of producing MP3 players over a two-year period is modeled by

where t is the time (in months). Approximate the average cost per unit over the two-year period.

Solution:

The average cost can be found by integrating c over the interval [0, 24].

2929

Example 4 – Solutioncont’d

Figure 5.12

3030

Even and Odd FunctionsExample 8: Evaluate each definite integral.

3131

Example 8 – Integrating Even and Odd Functions

Solution:

a. Because is an even function,

b. Because is an odd function,

3232

Closure: Applying the Definition of Average (Mean) Value

Find the average value of f (x) = 6 – x2 on [0, 5].

b

a

dxxfab

fav1

5

0

22 605

16 dxxxav

667.115

1

3334.2

11 warm-up: (let h be measured in feet) h(t) = -5t 2 + 20t + 15 a. estimate the instantaneous...

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