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### lecture18_handout

Course: MATH 1431, Fall 2008
School: U. Houston
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Word Count: 745

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18Section Lecture 5.5 Some Area Problems Jiwen He Quiz 1 What is today? a. b. c. d. Monday Wednesday Friday None of these 1 1.1 Section 5.5 Some Area Problems Area below the graph of a Nonnegative f Area below the graph of a Nonnegative f f (x) 0 for all x in [a, b]. = region below the graph of f . 1 b Area of = a f (x) dx = F (b) F (a) where F (x) is an antiderivative of f (x). Fundamental Theorem of...

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18Section Lecture 5.5 Some Area Problems Jiwen He Quiz 1 What is today? a. b. c. d. Monday Wednesday Friday None of these 1 1.1 Section 5.5 Some Area Problems Area below the graph of a Nonnegative f Area below the graph of a Nonnegative f f (x) 0 for all x in [a, b]. = region below the graph of f . 1 b Area of = a f (x) dx = F (b) F (a) where F (x) is an antiderivative of f (x). Fundamental Theorem of Integral Calculus Theorem 1. In general, b f (x) dx = F (b) F (a). a where F (x) is an antiderivative of f (x). 2 Quiz 2 1 Give the value of 1 x3 2x2 + sin(x) dx. 1 2 4 3 4 3 1 2 a. b. c. d. e. None of these Example 1 Example 2. Find the area below the graph of the square-root function from x = 0 to x = 1. 3 Example 2 Example 3. Find the area bounded above by the curve y = 4 x2 and below by the x-axis. Quiz 3 Give the area bounded between the x-axis and the graph of y = x2 + 1 for 4 1 x 2. a. b. c. d. e. 5 4 3 2 None of these 1.2 Area between the graphs of f and g Area between the graphs of two Nonnegative f and g f (x) g(x) 0 for all x in [a, b]. = region between the graphs of f (Top) and g (Bottom). b b Area of = a Top Bottom dx = a f (x) g(x) dx. Example 3 Example 4. Find the area bounded above by y = x + 2 and below by y = x2 . 5 Area between the graphs of f and g f (x) g(x) for all x in [a, b]. = region between the graphs of f (Top) and g (Bottom). b b Area of = a Top Bottom dx = a f (x) g(x) dx. Example 4 Example 5. Find the area of the region shown in the gure below. 6 Example 5 Example 6. Find the area between y = 4x and y = x3 from x = 2 to x = 2. Example 6 Example 7. Use integrals to represent the area of the region = 1 2 shaded in the gure below. 7 1.3 c a Signed Area f (x) as dx Signed Area f (x) 0 b for all x in [a, b] f (x) dx = Area of 1 a f (x) 0 c b for all x in [b, c] f (x) dx = Area of 2 8 c b c f (x) dx = a a f (x) dx + b f (x) dx = Area of 1 Area of 2 = Area above the x-axis Area below the x-axis. b a f (x) dx as Signed Area 9 b c d e b f (x) dx = a a f (x) dx + c f (x) dx + d f (x) dx + e f (x) dx = Area of 1 Area of 2 + Area of 3 Area of 4 = Area of 1 + Area of 3 Area of 2 + Area of 4 = Area above the x-axis Area below the x-axis. Example 7 Example 8. Evaluate 3 x2 2x dx and interpret the result in terms of areas. 1 Example 8 Example 9. Use integrals to represent the area of the region shaded in the gure below. 10 Quiz 4 The graph of y = f (x) is shown below. 1 has area 4 , 2 has area 4 , and 3 3 3 4 3 has area 3 . Give f (x) dx. 1 a. b. c. d. e. 0 4 3 8 3 4 None of these Quiz 5 4 The graph of y = f (x) is shown below. 1 has area 3 , 2 has area 4 , and 3 2 4 3 has area 3 . Give f (x) dx. 1 11 a. b. c. d. e. 0 4 3 8 3 4 None of these Quiz 6 4 The graph of y ...

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Even-Numbered Answers to Exercise Set 3.4 Quadratic Functions2. (a) Vertex:( 4, 5)6.(a) Vertex:( 5, 25)(b) The parabola opens upward. (c) y-intercept: 21 (d) Axis of symmetry: x = 4 (e)30 27 24 21 18 15 12 9 6 3 4 2 3 6 2 4 6 8 10(b) T
U. Houston - MATH - 1300
Exercise Set 2.4: Equations of LinesWrite an equation in slope-intercept form for each of the following lines. 1.y2For each of the following equations, (a) Write the equation in slope-intercept form. (b) Identify the slope and the y-intercept of
U. Houston - MATH - 1300
Odd-Numbered Answers to Exercise Set 4.4: Using Factoring to Solve Equations1. 3. 5. 7. 9.x = 7, x = 3 x = 2, x = 6 x = 5, x = 7 x = 18, x = 435. (a) x-intercept: 4 (b)( 4, 0 ) ( 0, 16 )(c) y-intercept: 16 Coordinates of y-intercept: (d) Vert