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C5 HW 2 of 3 - Integration (17931569)
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1. -Question Details LarCalcET7 5.2.005.MI. [4056297]
Find the sum by adding each term together. Use the summation capabilities of a graphing utility to verify your result.
6
(2i + 3)i = 1
2. -Question Details LarCalcET7 5.2.011. [4056909]
Use sigma notation to write the sum.
+ + + . . . + 19(1)
19(2)
19(3)
19(14)
i = 1
1
3. -Question Details LarCalcET7 5.2.029.MI. [4057124]
Use left and right endpoints and the given number of rectangles to find two approximations of the area of the region between the graph of the function and the x-axis over the given interval.
< Area <
f(x) = 2x + 1, [0, 2], 4 rectangles
4. -Question Details LarCalcET7 5.2.033. [4056944]
Use left and right endpoints and the given number of rectangles to find two approximations of the area of the region between the graph of the function and the x-axis over the given interval. (Round your answers to four decimal places.)
< Area <
f(x) = cos(x), 0, , 4 rectangles𝜋2
5. -Question Details LarCalcET7 5.2.063. [4056323]
Use the Midpoint Rule with n = 4 to approximate the area of the region bounded by the graph of the function and the x-axis over the given interval.
f(x) = x + 4, [0, 2]2
6. -Question Details LarCalcET7 5.3.016. [4056908]
Write a definite integral that yields the area of the region. (Do not evaluate the integral.)
f(x) = 6 − 3x
dx 0
7. -Question Details LarCalcET7 5.3.017. [4056602]
Write a definite integral that yields the area of the region. (Do not evaluate the integral.)
f(x) = 4 − |x|
dx -4
8. -Question Details LarCalcET7 5.3.019. [4057213]
Write a definite integral that yields the area of the region. (Do not evaluate the integral.)
f(x) = 25 − x2
dx -3
9. -Question Details LarCalcET7 5.3.021. [4056895]
Write a definite integral that yields the area of the region. (Do not evaluate the integral.)
f(x) = cos(x)
dx
0
10. -Question Details LarCalcET7 5.3.023. [4056949]
Write a definite integral that yields the area of the region. (Do not evaluate the integral.)
g(y) = y3
dy 0
11. -Question Details LarCalcET7 5.3.024. [4056671]
Write a definite integral that yields the area of the region. (Do not evaluate the integral.)
f(y) = (y − 2)2
dy 0
12. -Question Details LarCalcET7 5.3.027. [4056875]
Sketch the region whose area is given by the definite integral.
Use a geometric formula to evaluate the integral.
dx86
0
13. -Question Details LarCalcET7 5.3.029. [4101374]
Sketch the region whose area is given by the definite integral.
Use a geometric formula to evaluate the integral.
x dx4
0
14. -Question Details LarCalcET7 5.3.031. [4056313]
Sketch the region whose area is given by the definite integral.
Use a geometric formula to evaluate the integral.
(3x + 3) dx4
0
15. -Question Details LarCalcET7 5.3.033. [4056719]
Sketch the region whose area is given by the definite integral.
Use a geometric formula to evaluate the integral.
dx3 − |x|3
−3
16. -Question Details LarCalcET7 5.3.035.MI. [4056524]
Sketch the region whose area is given by the definite integral.
Then use a geometric formula to evaluate the integral (a > 0, r > 0).
dx4
16 − x2−4
17. -Question Details LarCalcET7 5.3.037. [4057192]
Evaluate the integral using the following values.
dx = 320, dx = 16, = 4x6
32
x6
2dx
6
2
x dx2
6
3
18. -Question Details LarCalcET7 5.3.038. [4056232]
Evaluate the integral using the following values.
x dx = 1,020, x dx = 30, dx = 68
2
3 8
2
8
2
x dx2
2
19. -Question Details LarCalcET7 5.3.039. [4056651]
Evaluate the integral using the following values.
x dx = 320, x dx = 16, dx = 46
2
3 6
2
6
2
x dx6
2
15
3
20. -Question Details LarCalcET7 5.3.041. [4057175]
Evaluate the integral using the following values.
x dx = 320, x dx = 16, dx = 46
2
3 6
2
6
2
(x − 14) dx6
2
21. -Question Details LarCalcET7 5.3.045. [4056411]
Given evaluate
(a)
(b)
(c)
(d)
and ,f(x) dx = 105
0f(x) dx = 2
7
5
f(x) dx.7
0
f(x) dx.0
5
f(x) dx.5
5
2f(x) dx.5
0
22. -Question Details LarCalcET7 5.3.044. [4056752]
Evaluate the integral using the following values.
x dx = 260, x dx = 10, dx = 26
4
3 6
4
6
4
(21 − 7x − x ) dx6
4
3
23. -Question Details LarCalcET7 5.3.047. [4056903]
(a)
(b)
(c)
(d)
Given dx = 13 and dx = -5, evaluate the following.f(x)8
3g(x)
8
3
[f(x) + g(x)] dx8
3
[g(x) − f(x)] dx8
3
2g(x) dx8
3
3f(x) dx8
3
24. -Question Details LarCalcET7 5.3.049. [4056484]
Use the table of values to find lower and upper estimates of
Assume that f is a decreasing function.lower estimate upper estimate
x 0 2 4 6 8 10
f(x)
.f(x) dx10
0
32 22 5 −10 −18 −35
25. -Question Details LarCalcET7 5.3.068. [4101253]
Find possible values of a and b that make the statement true.
cos(x) dx = 0b
a
(a, b) =
26. -Question Details LarCalcET7 5.4.006. [4056723]
Use a graphing utility to graph the integrand.
Use the graph to determine whether the definite integral is positive, negative, or zero.
sin x dx𝜋
0
positive
negative
zero
27. -Question Details LarCalcET7 5.4.009. [4101356]
Evaluate the definite integral. Use a graphing utility to verify your result.
(2x − 1) dx0
−1
28. -Question Details LarCalcET7 5.4.015. [4056774]
Evaluate the definite integral. Use a graphing utility to verify your result.
du4 u − 8
u1
29. -Question Details LarCalcET7 5.4.017. [4057162]
Evaluate the definite integral. Use a graphing utility to verify your result.
− 2 dt1
t3
−1
30. -Question Details LarCalcET7 5.4.019.MI. [4057185]
Evaluate the definite integral. Use a graphing utility to verify your result.
(t − t ) dt0
1/3 2/3−1
31. -Question Details LarCalcET7 5.4.023.MI. [4056642]
Evaluate the definite integral. Use a graphing utility to verify your result.
(sin(x) − 2) dx𝜋
0
32. -Question Details LarCalcET7 5.4.031. [4059378]
Evaluate the definite integral. Use a graphing utility to verify your result.
(2 + 4) dx4
x0
33. -Question Details LarCalcET7 5.4.036. [4056385]
Determine the area of the given region under the curve.
y = 1
x4
34. -Question Details LarCalcET7 5.4.037. [4056929]
Determine the area of the given region.
y = sin x
35. -Question Details LarCalcET7 5.4.039. [4056451]
Find the area of the region bounded by the graphs of the equations.
y = 7x + 5, x = 0, x = 2, y = 02
36. -Question Details LarCalcET7 5.4.043. [4059466]
Find the area of the region bounded by the graphs of the equations.
y = , x = 1, x = e, y = 03x
37. -Question Details LarCalcET7 5.4.044. [4059517]
Find the area of the region bounded by the graphs of the equations.
y = e , x = 0, x = 6, y = 0x
38. -Question Details LarCalcET7 5.4.063. [4056535]
Find F as a function of x and evaluate it at x = 0, x = 𝜋/6, and x = 𝜋/2.
=
=
=
=
F(x) = cos 𝜃 d𝜃x
0
F(x)
F(0)
F 𝜋6
F 𝜋2
39. -Question Details LarCalcET7 5.4.065. [4059300]
Let
where f is the function whose graph is shown in the figure.
(a) Estimate g(0), g(2), g(4), g(6), and g(8). g(0) =
g(2) =
g(4) =
g(6) =
g(8) =
(b) Find the largest open interval on which g is increasing. (Enter your answer using interval notation.)
Find the largest open interval on which g is decreasing. (Enter your answer using interval notation.)
(c) Identify any extrema of g.
g has a ---Select--- of at
(d) Sketch a rough graph of g.
g(x) = f(t) dtx
0
x = .
40. -Question Details LarCalcET7 5.4.066. [4056272]
Let
where f is the function whose graph is shown in the figure.
(a) Estimate g(0), g(2), g(4), g(6), and g(8).
g(0) =
g(2) =
g(4) =
g(6) =
g(8) =
(b) Find the largest open interval on which g is increasing. (Enter your answer using interval notation.)
Find the largest open interval on which g is decreasing. (Enter your answer using interval notation.)
(c) Identify any extrema of g.
g has a ---Select--- of at x = .
(d) Sketch a rough graph of g.
g(x) = f(t) dtx
0
Assignment Details
41. -Question Details LarCalcET7 5.4.067. [4056841]
Consider the following.
(a) Integrate to find F as a function of x.
F(x) =
(b) Demonstrate the Second Fundamental Theorem of Calculus by differentiating the result in part (a).
F'(x) =
F(x) = dt(t + 4)x
0
42. -Question Details LarCalcET7 5.4.074. [4059536]
Consider the following.
(a) Integrate to find F as a function of x.
(b) Demonstrate the Second Fundamental Theorem of Calculus by differentiating the result in Part (a).
F(x) = dtx 3t1
F(x) =
F '(x) =
43. -Question Details LarCalcET7 5.4.090. [4057040]
At 1:00 P.M., oil begins leaking from a tank at a rate of (7 + 0.8t) gallons per hour. (Round your answers to three decimal places.)
(a) How much oil is lost from 1:00 P.M. to 4:00 P.M.? gal
(b) How much oil is lost from 4:00 P.M. to 7:00 P.M.?
gal (c) Compare your answers from parts (a) and (b). What do you notice?
The second answer is ---Select--- because the rate of flow is ---Select--- .
44. -Question Details LarCalcET7 5.4.091. [4057105]
The graph shows the velocity, in feet per second, of a car accelerating from rest. Use the graph to estimate the distance the car travels in 8 seconds. ft
45. -Question Details LarCalcET7 5.4.103. [4056449]
A particle is moving along the x-axis. The position of the particle at time t is given by
Find the total distance the particle travels in 9 units of time. units
x(t) = t − 6t + 9t − 2, 0 ≤ t ≤ 9.3 2