1. 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) = = -10 ft/s a(t) = -10 ft/ s2 0 = -10t + 20 t = 2 h(2) = 35 feet

2. Area and the Fundamental Theorem of Calculus
5.4
Area and the Fundamental
Theorem of Calculus
Day 1 Homework: p. 348/9: 2-8 EVEN, EVEN
Day 2 Homework: p. 348: EVEN, p. 349: EVEN
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3. Objectives
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.

4. Rules for Definite Integrals
Order of Integration
Zero
Constant Multiple

5. Example 1 Using the Rules for Definite Integrals
Suppose Find each of the following integrals, if possible.

6. Example 1 Using the Rules for Definite Integrals
Suppose Find each of the following integrals, if possible.

7. Example 1 Using the Rules for Definite Integrals
Suppose Find each of the following integrals, if possible.
Not possible; not enough information given.

8. Area and Definite Integrals
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.
Figure 5.5

9. Example 2: Use the graph of the integrand and areas to evaluate the integral5 2 6
A=1/2(6)(2+5)= 21

10. 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 the line x = 2, as shown in Figure 5.6.
Figure 5.6

11. Example 3 – Evaluating a Definite Integral Using a Geometric Formula
cont’d
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

12. The Fundamental Theorem of Calculus
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.

13. Example 4: Evaluating a Definite Integral
−𝟏 𝟎 𝒙 𝟏/𝟑 + 𝒙 𝟐/𝟑 dx
𝟎 𝟑 𝒙−𝟐 𝟑 dx
No substitution needed here.
𝑥 4/3 4/3 + 𝑥 5/3 5/ −1
3𝑥 4/ 𝑥 5/ −1
3(0) 4/ (0) 5/3 5 = 0
3(1) 4/ (1) 5/3 5 =
0 – =
u = x – 2 du = dx −𝟐 𝟏 𝒖 𝟑 du 𝑢 −2 (1) (−2) 4 4 = −15 4
u = 3 - 2
u = 0 - 2

14. Closure: Area Under a Curve
Determine the area under the curve over the interval [–4, 4].

15. 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 = = 12 b1 = v(8) =544 b2 = v(20) = 160 A = ½ (12)( ) = 4224 units

16. Warm-Up, p. 349: 32
0 2 𝑥 1+ 2𝑥 2 𝑑𝑥 Let u = 1+ 2𝑥 2 du = 4x dx; 𝑑𝑢 4𝑥 =𝑑𝑥 u(2) = 1+ 2(2) 2 =9 and u(0) = 1+ 2(0) 2 = 𝑥 𝑢 ∙ 𝑑𝑢 4𝑥 = 𝑢 −1/2 du = 1 4 ∙ 𝑢 1/2 1/2 9 1 = 1 2 𝑢 1/ (9) 1/2 = 1.5 and 1 2 (1) 1/2 = = 1

17. Area and Definite Integrals

18. 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

19. Example 1 – Solution
cont’d
So, you can represent the area of the region by a definite integral. To find the area, use the Fundamental Theorem of Calculus.

20. Example 1 – Solution So, the area of the region is square units.
cont’d
So, the area of the region is square units.

21. Example 2 – Interpreting Absolute Value
Evaluate
Solution:
The region represented by the definite integral is shown in Figure 5.11.
Figure 5.11

22. Example 2 – Solution
cont’d
Using Property 3 of definite integrals, rewrite the integral as two definite integrals.

23. Marginal Analysis
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.

24. 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 to 101 units.
b. Find the change in profit when sales increase from to 110 units.

25. Example 3(a) – Solution
The change in profit obtained by increasing sales from 100 to 101 units is

26. Example 3(b) – Solution
The change in profit obtained by increasing sales from 100 to 110 units is

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

28. 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].

29. Example 4 – Solution
cont’d
Figure 5.12

30. Example 8: Evaluate each definite integral.
Even and Odd Functions

31. Example 8 – Integrating Even and Odd Functions
Solution:
a. Because is an even function,
b. Because is an odd function,

32. Closure: Applying the Definition of Average (Mean) Value
Find the average value of f (x) = 6 – x2 on [0, 5].