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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


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


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


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


Write a definite integral that yields the area of the region. (Do not evaluate the integral.)

f(x) = 6 − 3x

dx 0

javascript:doDetail('0')








f(x) = 4 − |x|

dx -4



f(x) = 25 − x2

dx -3



f(x) = cos(x)

dx

0






g(y) = y3

dy 0



f(y) = (y − 2)2

dy 0




Sketch the region whose area is given by the definite integral.

Use a geometric formula to evaluate the integral.

dx86

0





x dx4

0





(3x + 3) dx4

0





dx3 − |x|3

−3




Then use a geometric formula to evaluate the integral (a > 0, r > 0).

dx4

16 − x2−4


Evaluate the integral using the following values.

dx = 320, dx = 16, = 4x6

32

x6

2dx

6

2

x dx2

6

3



x dx = 1,020, x dx = 30, dx = 68

2

3 8

2

8

2

x dx2

2






x dx = 320, x dx = 16, dx = 46

2

3 6

2

6

2

x dx6

2

15

3



x dx = 320, x dx = 16, dx = 46

2

3 6

2

6

2

(x − 14) dx6

2


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



x dx = 260, x dx = 10, dx = 26

4

3 6

4

6

4

(21 − 7x − x ) dx6

4

3






(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


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


Find possible values of a and b that make the statement true.

cos(x) dx = 0b

a

(a, b) =





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


Evaluate the definite integral. Use a graphing utility to verify your result.

(2x − 1) dx0

−1



du4 u − 8

u1






− 2 dt1

t3

−1



(t − t ) dt0

1/3 2/3−1



(sin(x) − 2) dx𝜋

0



(2 + 4) dx4

x0


Determine the area of the given region under the curve.

y = 1

x4







Determine the area of the given region.

y = sin x


Find the area of the region bounded by the graphs of the equations.

y = 7x + 5, x = 0, x = 2, y = 02



y = , x = 1, x = e, y = 03x



y = e , x = 0, x = 6, y = 0x


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







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 = .



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


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


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) =


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--- .


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


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






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