Chapter 7 1. 1 The functions f ( x) = 2 − x 2 and g ( x) = − x are given. a. Find the area of the region bounded by the graphs of f (x ) and g (x) . b. Find the area of the region bounded by the graph of f (x) and the x-axis. c. Find the area of the region bounded by the graph of g (x) , the x-axis and the line x = 2 . Chapter 7 2 1. The functions f ( x) = 2 − x 2 and g ( x) = − x are given. a. Find the area of the region bounded by the graphs of f (x) and g (x) . +2: integrand +1: limits & answer 2 2 ∫ ⎡⎣(2 − x ) − ( − x)⎤⎦ dx = −1 2 ∫ (2 − x 2 −1 + x) dx = 2 b. x3 x2 9 2x − + = 3 2 −1 2 Find the area of the region bounded by the graph of f (x) and the x-axis. 2 x3 2 − x dx = 2 x − ( ) ∫ 3 − 2 2 +2: integral +1: answer 2 − 2 4 2 3 c. Find the area of the region bounded by the graph of g (x) , the x-axis and the line x = 2 . =4 2− 2 ∫ [0 − (− x )]dx = 2 0 +2 integral +1: answer Chapter 7 2. 3 The functions f ( x) = x 2 − 2 and g ( x) = x are given. a. Find the area of the region(s) bounded by the graphs of f (x) , the y-axis and the x-axis. b. Find the area of the region bounded by the graphs of f (x) and g (x) . c. Find the area of the region bounded by the graph of g (x) and the x-axis between the lines x = −2 and x = 1 . Chapter 7 4 The functions f ( x) = x 2 − 2 and g ( x) = x are given. 2. a. Find the area of the region(s) bounded by the graphs of f (x) , the y-axis and the x-axis. +2: integral +1: answer The area of the first region is given by 2 ∫ 0 ⎡0 − ( x 2 − 2 ) ⎤dx = ⎣ ⎦ 2 ∫ (−x 0 2 + 2 ) dx x3 = − + 2x 3 0 2 4 2 3 The second region, which is to the left of the y axis, has the same area. So the total 8 2 area is . 3 = b. Find the area of the region bounded by the graphs of f ( x) and g ( x) . 2 ∫ [x − (x −1 2 +2: integral +1: answer ] − 2) dx = 2 9 x2 x3 − + 2x = 2 3 2 −1 c. Find the area of the region bounded by the graph of g ( x) and the x-axis between the lines x = −2 and x = 1 . 0 1 −2 0 5 ∫ (0 − x )dx + ∫ xdx = 2 +2: integral +1: answer Chapter 7 5 y 3. x F ( y ) = ∫ 3e 2 dx −1 a. Find the accumulation function F. b. Evaluate F(-1), F(0), and F(4). c. Graphically show the area given by the value F(0). Chapter 7 6 y 3. x F ( y ) = ∫ 3e 2 dx −1 a. Find the accumulation function F. +2: antiderivative y ∫ 3e 2 dx = 6e x x y 2 y b. +1: answer −1 −1 −1 = 6e 2 − 6e 2 Evaluate F(-1), F(0), and F(4). F (−1) = 0 +1 for 0 F ( 0) = 6 − 6e 2 −1 +1 for 6 − 6e 2 −1 F ( 4) = 6e − 6e c. Graphically show the area given by the value F(0). 2 −1 2 +1 for 6e 2 − 6e −1 +2: graph of 3e +1: shaded area x 2 2 Chapter 7 7 x 4. F ( x) = ∫ (2t + 1)dt 0 a. Find the accumulation function F. b. Evaluate F(0), F(2), and F(6). c. Graphically illustrate the area given by the value F(2). Chapter 7 8 x 4. F ( x) = ∫ (2t + 1)dt 0 a. Find the accumulation function F. +2: antiderivative x ∫ (2t + 1)dt = +1: answer 0 x t 2 + t = x2 + x 0 b. Evaluate F(0), F(2), and F(6). F (0) = 0 F (2) = 6 F (6) = 42 c. Graphically illustrate the area given by the value F(2). +1: 0 +1: 6 +1: 42 +2: graph of y = 2t + 1 +1: shaded area Chapter 7 5. 9 Consider the region bounded by the graphs of f ( x) = x , y = 0 , and x = 2 . a. Find the volume of the solid formed by rotating the region about the xaxis. b. Find the volume of the solid formed by rotating the region about the yaxis. c. Find the volume of the solid formed by rotating the region about the line y = −2 . Chapter 7 5. 10 Consider the region bounded by the graphs of f ( x) = x , y = 0 , and x = 2 . a. Find the volume of the solid formed by rotating the region about the x-axis. +1: limits, constant, answer ( x ) dx = 2 2 π∫ 0 x2 π 2 +2: integral 2 = 2π ≈ 6.283 0 b. Find the volume of the solid formed by rotating the region about the y-axis. +2: integral 2 π ∫ (4 − y )dy = 4 +1: limits, constant, answer 0 2 ⎛ y5 ⎞ 16 2 π ⎜⎜ 4 y − ⎟⎟ = π ≈ 14.217 5 ⎠0 5 ⎝ c. Find the volume of the solid formed by rotating the region about the line y = −2 . ( 2 ) π ∫ ⎡⎢ x + 2 − 4⎤⎥ dx = ⎣ ⎦ 0 2 2 ( ) π ∫ x + 4 x + 4 − 4 dx = 0 2 ⎛ x2 8 3 ⎞ π ⎜⎜ + x 2 ⎟⎟ = ⎠0 ⎝ 2 3 16 ⎛ ⎞ 2 ⎟π ≈ 29.979 ⎜2 + 3 ⎝ ⎠ +2: integral +1: limits, constant, answer Chapter 7 6. 11 Consider the region bounded by y = x 3 , y = 8 , and x = 0 . a. Find the volume of the solid formed by rotating the region about the yaxis. b. Find the volume of the solid formed by rotating the region about the xaxis. c. Find the volume of the solid formed by rotating the region about the line x = 2. Chapter 7 6. 12 Consider the region bounded by y = x 3 , y = 8 , and x = 0 . . a. Find the volume of the solid formed by rotating the region about the y-axis +2: integrand 8 π∫ 0 +1: limits, constant, answer ( y ) dy = 3 2 8 96 ⎛3 5 ⎞ π ⎜ y 3 ⎟ = π ≈ 60.319 5 ⎝5 ⎠0 b. Find the volume of the solid formed by rotating the +2: integrand region about the x-axis. 2 ( ) π ∫ 82 − ( x 3 ) dx = 0 2 +1: limits, constant, answer 2 π ∫ ( 64 − x 6 ) dx = 0 2 ⎡ x7 ⎤ 768 π ⎢64 x − ⎥ = π ≈ 344.678 7 ⎦0 7 ⎣ c. Find the volume of the solid formed by rotating the +2: integrand region about the line x = 2 . ( 8 )⎠ π ∫ ⎛⎜ 2 2 − 2 − 3 y ⎞⎟dy = 0 ⎝ 8 π ∫ ⎛⎜ 4 y ⎝ 0 1 3 2 −y 2 3 ⎞⎟dy = ⎠ 8 3 5 ⎤ ⎡ 4 π ⎢3 y 3 − y 3 ⎥ = 5 ⎣ ⎦0 ⎡ ⎣ 3 5 ⎤ ⎦ π ⎢3(16) − (32)⎥ = 144 π ≈ 90.478 5 +1: limits, constant, answer Chapter 7 7. 13 Do the following: a. Find the distance between the points (1, 2) and (7, 10) using integration. b. Find the length of the curve y = 1 + 6 x 2 on the interval [0, 1]. c. Find the length of the curve y = 3 x3 1 1 + , ≤ x ≤ 1. 6 2x 2 Chapter 7 14 7. Do the following: a. Find the distance between the points (1, 2) and (7, 10) using integration. +1: answer 2 7 ⎛4⎞ 1 + ⎜ ⎟ dx = ⎝3⎠ ∫ 1 +2: integral 7 7 5 5 ∫1 3 dx = 3 x 1 = 10 3 b. Find the length of the curve y = 1 + 6 x 2 on the interval [0, 1]. 1 1 + 9x 0 2 +1: answer 1 ( ) dx = ∫ ∫ 1 2 1 + 81x dx = 0 1 ⎞ ⎟ = 2 82 3 2 − 1 ⎟ ⎠ 0 243 c. Find the length of the curve 3 x 1 1 y= + , ≤ x ≤1 6 2x 2 1 ⎛⎜ 2(1 + 81x ) 81 ⎜⎝ 3 1 ∫ 1 2 1 ∫ 1 +2: integral 2 3 ( 2 ⎛ x2 1 1 + ⎜⎜ − 2 ⎝ 2 2x ⎛ x3 1 ⎞ ⎜⎜ − ⎟⎟ ⎝ 6 2x ⎠ = 1 2 +2: integral +1: answer 2 ⎞ ⎟⎟ dx = ⎠ x4 1 1 + + 4 dx = 4 2 4x 1 ) 31 48 1 ∫ 1 2 ⎛ x2 1 ⎜⎜ + 2 ⎝ 2 2x 2 ⎞ ⎟⎟ dx = . ⎠ Chapter 7 8. 15 Let R be the region bounded by the graph of y = ln x and the line y = 2 x − 3 . a. Find the area of R. b. Find the volume of the solid generated when R is rotated about the horizontal line y = −3 . c. Write, but do not evaluate, an integral expression that can be used to find the volume of the solid generated when R is rotated about the y-axis. Chapter 7 8. 16 Let R be the region bounded by the graph of y = ln x and the line y = 2 x − 3 . ln x = 2 x − 3 when x = 0.05565 and 1.79154. Let A = 0.05565 and B = 1.79154. a. Find the area of R. +1: integrand B Area = ∫ (ln x − (2 x − 3))dx = 1.471 +1: limits A +1: answer b. Find the volume of the solid generated when R is rotated about the horizontal line y = −3 . B ( ) Volume = π ∫ (ln x + 3) − (2 x ) dx = 18.783 A 2 2 c. Write, but do not evaluate, an integral expression that can be used to find the volume of the solid generated when R is rotated about the y-axis. Volume = π ⎛⎛ y + 3 ⎞2 ⎜⎜ − ey ∫ ⎜ ⎝ 2 ⎟⎠ 2 A−3 ⎝ 2 B −3 ⎞ ( ) ⎟⎟dy 2 ⎠ +2: integrand +1: limits, constant, answer +2: integrand +1: limits & constant Chapter 7 9. 17 Let R be the region bounded by the graph of y = x 2 − 1 and the graph of x = y 2 . a. Find the area of R. b. Find the volume of the solid generated when R is rotated about the vertical line x = 2 . c. Write, but do not evaluate, an integral expression that can be used to find the volume of the solid generated when R is rotated about the line y = −1 . Chapter 7 18 Let R be the region bounded by the graph of y = x 2 − 1 and the graph of x = y 2 . 9. x 2 −1 = ± x when x = 0.52489 and 1.49022. Let A = 0.52489 and B = 1.49022. Let S = − A = −0.72449 and T = 1.49022 = 1.22074 a. Find the area of R. ∫( Area = +1: integrand +1: limits +1: answer ) T y + 1 − y 2 dy = 1.377 S b. Find the volume of the solid generated when R is rotated about the vertical line x = 2 . ( T ) 2 2 V = π ∫ ⎛⎜ (2 − y 2 ) − 2 − y + 1 ⎞⎟dy = 11.501 ⎠ ⎝ S +2: integrand +1: limits, constant, answer c. Write, but do not evaluate, an integral expression that can be used to find the volume of the solid generated when R is rotated about the line y = −1 . +2: integrand V= +1: limits & constant A π∫ 0 (( ) ( 2 ) ) dx + π ∫ (( x +1 − − x +1 2 B A ) ) x + 1 − ( x 2 ) dx 2 2 Chapter 7 19 10. a. Neglecting air resistance and the weight of the propellant, determine the work done in propelling an 8-ton satellite to a height of 200 miles above Earth. (Use 4000 miles as the radius of Earth.) b. A lunar module weighs 15 tons on the surface of Earth. How much work is done in propelling the module from the surface of the moon to a height of 100 miles? Consider the radius of the moon to be 1100 miles and its force of gravity to be one-sixth that of Earth. c. Find the center of mass of a system of point masses m1 = 4 , m2 = 6 , m3 = 2 , and m 4 = 8 , located at (3, -2), (1, -1), (-4,6), and (4, 2), respectively. Chapter 7 20 10. a. Neglecting air resistance and the weight of the propellant, determine the work done in propelling an 8-ton satellite to a height of 200 miles above Earth. (Use 4000 +2: integrand miles as the radius of Earth.) +1: limits, answer 4200 128,000,000 dx = ∫ x2 4000 4200 − 128000000 = 1523.810 mile − tons x 4000 b. A lunar module weighs 15 tons on the surface of Earth. How much work is done in propelling the module from the surface of the moon to a height of 100 miles? Consider the radius of the moon to be 1100 miles and its force of gravity to be one-sixth that of Earth. +2: integrand +1: limits, answer 1200 3,025,000 dx = 2 x 1100 ∫ 1200 − 3025000 = 229.167 mile − tons x 1100 c. Find the center of mass of a system of point masses m1 = 4 , m 2 = 6 , m3 = 2 , and m 4 = 8 , located at (3, -2), (1, -1), (-4,6), and (4, 2), respectively. m = 6 + 2 + 8 + 4 = 20 M y = 4(3) + 6(1) + 2(−4) + 8(4) = 42 M x = 4(−2) + 6(−1) + 2(6) + 8(2) = 14 ⎛ 21 7 ⎞ Center of mass is ⎜ , ⎟ ⎝ 10 10 ⎠ +1: M y +1: M x +1: center of mass