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

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, 18-28 EVEN Day 2 Homework: p. 348: 10-16 EVEN, p. 349: 48-56 EVEN Copyright © Cengage Learning. All rights reserved.

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.

Rules for Definite Integrals Order of Integration Zero Constant Multiple

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

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

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

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

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

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

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

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.

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 5/3 0 −1 3𝑥 4/3 4 + 3𝑥 5/3 5 0 −1 3(0) 4/3 4 + 3(0) 5/3 5 = 0 3(1) 4/3 4 + 3(1) 5/3 5 = 27 20 0 – 27 20 = - 27 20 u = x – 2 du = dx −𝟐 𝟏 𝒖 𝟑 du 𝑢 4 4 1 −2 (1) 4 4 - (−2) 4 4 = −15 4 u = 3 - 2 u = 0 - 2

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

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

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 = 1 1 9 𝑥 𝑢 ∙ 𝑑𝑢 4𝑥 = 1 4 1 9 𝑢 −1/2 du = 1 4 ∙ 𝑢 1/2 1/2 9 1 = 1 2 𝑢 1/2 9 1 1 2 (9) 1/2 = 1.5 and 1 2 (1) 1/2 = .5 1.5 - .5 = 1

Area and Definite Integrals

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

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.

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

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

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

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.

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.

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

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

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

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].

Example 4 – Solution cont’d Figure 5.12

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

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

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