4 Approximating with a Riemann sum: Use & with 10 rectangles to approximate the a graph of f(x)=x² on (0.5) Limits of Riemann sums: Use a limit of Raman to calculate the exact area under the ga on [0,5]. responding position function. Explain why this suggests that the signed area under the velocity an interval is equal to the difference in the positi tion on that interval, and tell what this has to do Fundamental Theorem of Calculus 8. Why is it not necessary to write down an antib family when using the Fundamental Theorem of C to calculate definite integrals? In other words, wh we have to use "+C"? Suppose f is a function whose derivative f' is gb graph shown next. In Exercises 9-12, use the g off, an area approximation, and the Fundamental The to approximate the requested value. Graph of f', the derivative off 3. 16. Determine whether or not each statement that follows is equivalent to the Fundamental Theorem of Calculus. Assume that all functions here are integrable. (a) f(x)=F(x), then F(x) dx = [f(x)] (b) G(x) dx = (G'(x) (c) If h(x) is the derivative of g(x), then h(x) dx = 8(b)-8(a). (d) [fwx) dx-(x) dr. (e) [fs'(x) dx]=[s" (x)) 17. Determine whether or not each statement that follows is equivalent to the Fundamental Theorem of Calculus. Assume that all functions here are integrable. Skills Use the Fundamental Theorem of Calculus to find the exact values of each of the definite integrals in Exercises 19-64. Use a graph to check your answer. (Hint: The integrands that involve absolute values will have to be considered piecewise.) 4.5 The Fundamental Theorem of Calculus (a) f(x) dxg(b)-g(a), where g'(x) = f(x). (b) If f'(x) = F(x), then f(x) dx = [F(x) (c) If g(x) is any antiderivative of h(x), then Sh(x) dx = g(a)-8(b). (d) f(x) dx = ['(x) (e) [p(x) dx] = p(x) dx. 18. In the proof of the Fundamental Theorem of Cal- culus, the Mean Value Theorem is used to choose val- ues x in each subinterval [x-1.x]. Use the Mean Value Theorem in the same way to find the corresponding values x; for a Riemann sum approximation of x2dr with four rectangles. x²+5dx 2x 42. dx 2x 41. x²+5 2x+3 dx 44. Cscxcotxdr 43. x2+3x+4 J/4 19. (x+3x+1) dr 20. La- (x-1)(x+3)dx dx 46. 200 2 cos(xx) dx 45. -3 3л 21. 3(2) dr 22. (x²-4)² dx dx 48. 2(lnx) ( dx 47. 2x+6 1 23. dx 24. sin(3x) dx 49. (x2/3-2/3) dx 50. Jo (2x+1)5/2 dx 25. √-1dx 1/2 Inx 26. 2x√√x²+1dx 51. sin x(1+ cos x) dx 52. dx x 27. secx tanx dx 28. -J sec² x dx ex-xex x²ex-2xex 53. dx 54. dx √2/4 1 1/2 C=/2 29. 30. 1-4x 1+ 55. xcsc²(x2) dx 56. x² sec²(x³) dx J/4 -/4 31. + sin x) dx 32. 32x-4 dx 3x 2x-3x 57. dx 58. dx 1-x2 4* 33. 2 35. 34. 4-3x 59. x-21 dx 60. -2 L 3 14-x2|dx -1 36. S dx 1+x2 61. 12x²-5x-31 dx 62. 37. dr 38. (sin² x + cos² x) dx 5/4 63. |sin x) dx 64. (x-1)(x-4) dx 2x-11 dx 39. dx 40. dx (x+5)2 9. Given that f(3) = 2, approximate f(4). 10. Given that f(0)=-1, approximate f(2). 11. Given that f(2) = 3, approximate f(-2). 12. Given that f(-1)=2, approximate f(1). Calculate each definite integral in Exercises 13-14, using the definition of the definite integral as a limit of Ri sums, (b) the definite integral formulas from Theorem and (c) the Fundamental Theorem of Calculus. Then so that your three answers are the same. 13. 3x² 3x2 dx 14. (3x+2) 15. In the proof of the Fundamental Theorem of Calculs encounter a telescoping sum. Find the values of the lowing sums, which are also telescoping. (a) 100 EG-) 100 kml Applications 65. Suppose a large oil tank develops a hole that causes oil to leak from the tank at a rate of r(t) = 0.05t gallons per hour. Construct and then solve a definite integral to determine the amount of oil that has leaked from the tank after one day. 66. Suppose another oil tank with a hole in it takes three days to leak 11 gallons of oil. Assuming that the rate of leakage at time t is given by r(t)=kt for some constant k set up and solve an equation involving a definite integra to find k.
4 Approximating with a Riemann sum: Use & with 10 rectangles to approximate the a graph of f(x)=x² on (0.5) Limits of Riemann sums: Use a limit of Raman to calculate the exact area under the ga on [0,5]. responding position function. Explain why this suggests that the signed area under the velocity an interval is equal to the difference in the positi tion on that interval, and tell what this has to do Fundamental Theorem of Calculus 8. Why is it not necessary to write down an antib family when using the Fundamental Theorem of C to calculate definite integrals? In other words, wh we have to use "+C"? Suppose f is a function whose derivative f' is gb graph shown next. In Exercises 9-12, use the g off, an area approximation, and the Fundamental The to approximate the requested value. Graph of f', the derivative off 3. 16. Determine whether or not each statement that follows is equivalent to the Fundamental Theorem of Calculus. Assume that all functions here are integrable. (a) f(x)=F(x), then F(x) dx = [f(x)] (b) G(x) dx = (G'(x) (c) If h(x) is the derivative of g(x), then h(x) dx = 8(b)-8(a). (d) [fwx) dx-(x) dr. (e) [fs'(x) dx]=[s" (x)) 17. Determine whether or not each statement that follows is equivalent to the Fundamental Theorem of Calculus. Assume that all functions here are integrable. Skills Use the Fundamental Theorem of Calculus to find the exact values of each of the definite integrals in Exercises 19-64. Use a graph to check your answer. (Hint: The integrands that involve absolute values will have to be considered piecewise.) 4.5 The Fundamental Theorem of Calculus (a) f(x) dxg(b)-g(a), where g'(x) = f(x). (b) If f'(x) = F(x), then f(x) dx = [F(x) (c) If g(x) is any antiderivative of h(x), then Sh(x) dx = g(a)-8(b). (d) f(x) dx = ['(x) (e) [p(x) dx] = p(x) dx. 18. In the proof of the Fundamental Theorem of Cal- culus, the Mean Value Theorem is used to choose val- ues x in each subinterval [x-1.x]. Use the Mean Value Theorem in the same way to find the corresponding values x; for a Riemann sum approximation of x2dr with four rectangles. x²+5dx 2x 42. dx 2x 41. x²+5 2x+3 dx 44. Cscxcotxdr 43. x2+3x+4 J/4 19. (x+3x+1) dr 20. La- (x-1)(x+3)dx dx 46. 200 2 cos(xx) dx 45. -3 3л 21. 3(2) dr 22. (x²-4)² dx dx 48. 2(lnx) ( dx 47. 2x+6 1 23. dx 24. sin(3x) dx 49. (x2/3-2/3) dx 50. Jo (2x+1)5/2 dx 25. √-1dx 1/2 Inx 26. 2x√√x²+1dx 51. sin x(1+ cos x) dx 52. dx x 27. secx tanx dx 28. -J sec² x dx ex-xex x²ex-2xex 53. dx 54. dx √2/4 1 1/2 C=/2 29. 30. 1-4x 1+ 55. xcsc²(x2) dx 56. x² sec²(x³) dx J/4 -/4 31. + sin x) dx 32. 32x-4 dx 3x 2x-3x 57. dx 58. dx 1-x2 4* 33. 2 35. 34. 4-3x 59. x-21 dx 60. -2 L 3 14-x2|dx -1 36. S dx 1+x2 61. 12x²-5x-31 dx 62. 37. dr 38. (sin² x + cos² x) dx 5/4 63. |sin x) dx 64. (x-1)(x-4) dx 2x-11 dx 39. dx 40. dx (x+5)2 9. Given that f(3) = 2, approximate f(4). 10. Given that f(0)=-1, approximate f(2). 11. Given that f(2) = 3, approximate f(-2). 12. Given that f(-1)=2, approximate f(1). Calculate each definite integral in Exercises 13-14, using the definition of the definite integral as a limit of Ri sums, (b) the definite integral formulas from Theorem and (c) the Fundamental Theorem of Calculus. Then so that your three answers are the same. 13. 3x² 3x2 dx 14. (3x+2) 15. In the proof of the Fundamental Theorem of Calculs encounter a telescoping sum. Find the values of the lowing sums, which are also telescoping. (a) 100 EG-) 100 kml Applications 65. Suppose a large oil tank develops a hole that causes oil to leak from the tank at a rate of r(t) = 0.05t gallons per hour. Construct and then solve a definite integral to determine the amount of oil that has leaked from the tank after one day. 66. Suppose another oil tank with a hole in it takes three days to leak 11 gallons of oil. Assuming that the rate of leakage at time t is given by r(t)=kt for some constant k set up and solve an equation involving a definite integra to find k.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.2: Graphs Of Equations
Problem 78E
Related questions
Question
100%
answer questions 13, 29, 33, 49, 59
![4
Approximating with a Riemann sum: Use &
with 10 rectangles to approximate the a
graph of f(x)=x² on (0.5)
Limits of Riemann sums: Use a limit of Raman
to calculate the exact area under the ga
on [0,5].
responding position function. Explain why this
suggests that the signed area under the velocity
an interval is equal to the difference in the positi
tion on that interval, and tell what this has to do
Fundamental Theorem of Calculus
8. Why is it not necessary to write down an antib
family when using the Fundamental Theorem of C
to calculate definite integrals? In other words, wh
we have to use "+C"?
Suppose f is a function whose derivative f' is gb
graph shown next. In Exercises 9-12, use the g
off, an area approximation, and the Fundamental The
to approximate the requested value.
Graph of f', the derivative off
3.
16. Determine whether or not each statement that follows
is equivalent to the Fundamental Theorem of Calculus.
Assume that all functions here are integrable.
(a) f(x)=F(x), then F(x) dx = [f(x)]
(b) G(x) dx = (G'(x)
(c) If h(x) is the derivative of g(x), then h(x) dx =
8(b)-8(a).
(d) [fwx) dx-(x) dr.
(e) [fs'(x) dx]=[s" (x))
17. Determine whether or not each statement that follows
is equivalent to the Fundamental Theorem of Calculus.
Assume that all functions here are integrable.
Skills
Use the Fundamental Theorem of Calculus to find the exact
values of each of the definite integrals in Exercises 19-64. Use
a graph to check your answer. (Hint: The integrands that involve
absolute values will have to be considered piecewise.)
4.5 The Fundamental Theorem of Calculus
(a) f(x) dxg(b)-g(a), where g'(x) = f(x).
(b) If f'(x) = F(x), then f(x) dx = [F(x)
(c) If g(x) is any antiderivative of h(x), then
Sh(x) dx = g(a)-8(b).
(d) f(x) dx = ['(x)
(e) [p(x) dx] = p(x) dx.
18. In the proof of the Fundamental Theorem of Cal-
culus, the Mean Value Theorem is used to choose val-
ues x in each subinterval [x-1.x]. Use the Mean Value
Theorem in the same way to find the corresponding
values x; for a Riemann sum approximation of x2dr
with four rectangles.
x²+5dx
2x
42.
dx
2x
41.
x²+5
2x+3
dx
44.
Cscxcotxdr
43.
x2+3x+4
J/4
19.
(x+3x+1) dr
20.
La-
(x-1)(x+3)dx
dx
46.
200
2 cos(xx) dx
45.
-3 3л
21.
3(2) dr
22.
(x²-4)² dx
dx
48.
2(lnx) (
dx
47.
2x+6
1
23.
dx
24.
sin(3x) dx
49.
(x2/3-2/3) dx
50.
Jo (2x+1)5/2
dx
25.
√-1dx
1/2
Inx
26.
2x√√x²+1dx
51.
sin x(1+ cos x) dx
52.
dx
x
27.
secx tanx dx
28.
-J
sec² x dx
ex-xex
x²ex-2xex
53.
dx
54.
dx
√2/4
1
1/2
C=/2
29.
30.
1-4x
1+
55.
xcsc²(x2) dx
56.
x² sec²(x³) dx
J/4
-/4
31.
+ sin x) dx
32.
32x-4 dx
3x
2x-3x
57.
dx
58.
dx
1-x2
4*
33.
2
35.
34.
4-3x
59.
x-21 dx
60.
-2
L
3
14-x2|dx
-1
36.
S
dx
1+x2
61. 12x²-5x-31 dx
62.
37.
dr
38.
(sin² x + cos² x) dx
5/4
63.
|sin x) dx
64.
(x-1)(x-4) dx
2x-11 dx
39.
dx
40.
dx
(x+5)2
9. Given that f(3) = 2, approximate f(4).
10. Given that f(0)=-1, approximate f(2).
11. Given that f(2) = 3, approximate f(-2).
12. Given that f(-1)=2, approximate f(1).
Calculate each definite integral in Exercises 13-14, using
the definition of the definite integral as a limit of Ri
sums, (b) the definite integral formulas from Theorem
and (c) the Fundamental Theorem of Calculus. Then so
that
your three answers are the same.
13.
3x²
3x2 dx
14.
(3x+2)
15. In the proof of the Fundamental Theorem of Calculs
encounter a telescoping sum. Find the values of the
lowing sums, which are also telescoping.
(a)
100
EG-)
100
kml
Applications
65. Suppose a large oil tank develops a hole that causes oil
to leak from the tank at a rate of r(t) = 0.05t gallons
per hour. Construct and then solve a definite integral to
determine the amount of oil that has leaked from the tank
after one day.
66. Suppose another oil tank with a hole in it takes three
days to leak 11 gallons of oil. Assuming that the rate of
leakage at time t is given by r(t)=kt for some constant k
set up and solve an equation involving a definite integra
to find k.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2d2a894b-5d23-4d74-addb-138aa024bb4a%2Faca9d5bb-24e7-41b3-8b6e-eac0c5d20d16%2Fruld6hw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:4
Approximating with a Riemann sum: Use &
with 10 rectangles to approximate the a
graph of f(x)=x² on (0.5)
Limits of Riemann sums: Use a limit of Raman
to calculate the exact area under the ga
on [0,5].
responding position function. Explain why this
suggests that the signed area under the velocity
an interval is equal to the difference in the positi
tion on that interval, and tell what this has to do
Fundamental Theorem of Calculus
8. Why is it not necessary to write down an antib
family when using the Fundamental Theorem of C
to calculate definite integrals? In other words, wh
we have to use "+C"?
Suppose f is a function whose derivative f' is gb
graph shown next. In Exercises 9-12, use the g
off, an area approximation, and the Fundamental The
to approximate the requested value.
Graph of f', the derivative off
3.
16. Determine whether or not each statement that follows
is equivalent to the Fundamental Theorem of Calculus.
Assume that all functions here are integrable.
(a) f(x)=F(x), then F(x) dx = [f(x)]
(b) G(x) dx = (G'(x)
(c) If h(x) is the derivative of g(x), then h(x) dx =
8(b)-8(a).
(d) [fwx) dx-(x) dr.
(e) [fs'(x) dx]=[s" (x))
17. Determine whether or not each statement that follows
is equivalent to the Fundamental Theorem of Calculus.
Assume that all functions here are integrable.
Skills
Use the Fundamental Theorem of Calculus to find the exact
values of each of the definite integrals in Exercises 19-64. Use
a graph to check your answer. (Hint: The integrands that involve
absolute values will have to be considered piecewise.)
4.5 The Fundamental Theorem of Calculus
(a) f(x) dxg(b)-g(a), where g'(x) = f(x).
(b) If f'(x) = F(x), then f(x) dx = [F(x)
(c) If g(x) is any antiderivative of h(x), then
Sh(x) dx = g(a)-8(b).
(d) f(x) dx = ['(x)
(e) [p(x) dx] = p(x) dx.
18. In the proof of the Fundamental Theorem of Cal-
culus, the Mean Value Theorem is used to choose val-
ues x in each subinterval [x-1.x]. Use the Mean Value
Theorem in the same way to find the corresponding
values x; for a Riemann sum approximation of x2dr
with four rectangles.
x²+5dx
2x
42.
dx
2x
41.
x²+5
2x+3
dx
44.
Cscxcotxdr
43.
x2+3x+4
J/4
19.
(x+3x+1) dr
20.
La-
(x-1)(x+3)dx
dx
46.
200
2 cos(xx) dx
45.
-3 3л
21.
3(2) dr
22.
(x²-4)² dx
dx
48.
2(lnx) (
dx
47.
2x+6
1
23.
dx
24.
sin(3x) dx
49.
(x2/3-2/3) dx
50.
Jo (2x+1)5/2
dx
25.
√-1dx
1/2
Inx
26.
2x√√x²+1dx
51.
sin x(1+ cos x) dx
52.
dx
x
27.
secx tanx dx
28.
-J
sec² x dx
ex-xex
x²ex-2xex
53.
dx
54.
dx
√2/4
1
1/2
C=/2
29.
30.
1-4x
1+
55.
xcsc²(x2) dx
56.
x² sec²(x³) dx
J/4
-/4
31.
+ sin x) dx
32.
32x-4 dx
3x
2x-3x
57.
dx
58.
dx
1-x2
4*
33.
2
35.
34.
4-3x
59.
x-21 dx
60.
-2
L
3
14-x2|dx
-1
36.
S
dx
1+x2
61. 12x²-5x-31 dx
62.
37.
dr
38.
(sin² x + cos² x) dx
5/4
63.
|sin x) dx
64.
(x-1)(x-4) dx
2x-11 dx
39.
dx
40.
dx
(x+5)2
9. Given that f(3) = 2, approximate f(4).
10. Given that f(0)=-1, approximate f(2).
11. Given that f(2) = 3, approximate f(-2).
12. Given that f(-1)=2, approximate f(1).
Calculate each definite integral in Exercises 13-14, using
the definition of the definite integral as a limit of Ri
sums, (b) the definite integral formulas from Theorem
and (c) the Fundamental Theorem of Calculus. Then so
that
your three answers are the same.
13.
3x²
3x2 dx
14.
(3x+2)
15. In the proof of the Fundamental Theorem of Calculs
encounter a telescoping sum. Find the values of the
lowing sums, which are also telescoping.
(a)
100
EG-)
100
kml
Applications
65. Suppose a large oil tank develops a hole that causes oil
to leak from the tank at a rate of r(t) = 0.05t gallons
per hour. Construct and then solve a definite integral to
determine the amount of oil that has leaked from the tank
after one day.
66. Suppose another oil tank with a hole in it takes three
days to leak 11 gallons of oil. Assuming that the rate of
leakage at time t is given by r(t)=kt for some constant k
set up and solve an equation involving a definite integra
to find k.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 4 images
Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage