Question 1

Use a &#64257;nite sum to estimate the average value of the function on the given interval by partitioning the interval and&#10;evaluating the function at the midpoints of the subintervals.&#10;

Accepted Answer

To estimate the average value of the function $f(t) = \frac{1}{e^{-t}} \cos 2\pi t$ on the interval $[-2,3]$ using 5 subintervals, we first calculate the width of each subinterval: $\Delta t = \frac{3 - (-2)}{5} = 1$. The midpoints of these subintervals are $-1.5, -0.5, 0.5, 1.5,$ and $2.5$. Evaluating the function at these midpoints and averaging the results gives an approximation of the average value of the function over the interval. The calculation involves substituting each midpoint into the function, summing these values, and then dividing by 5. The result, rounded to the nearest hundredth, is approximately -1.41.

Question 2

f(x) = x2 between x = 0 and x = 1 using the &#34;midpoint rule&#34; with two rectangles of equal width.&#10;A) )3145&#10;B) )625&#10;C) )75&#10;D) )125

Accepted Answer

The midpoint rule with two rectangles involves evaluating the function at the midpoints of the intervals. The interval [0,1] is divided into two subintervals: [0,0.5] and [0.5,1]. The midpoints are 0.25 and 0.75. The function values at these points are f(0.25) = 0.25^2 = 0.0625 and f(0.75) = 0.75^2 = 0.5625. The width of each rectangle is 0.5. The estimated area is 0.5*(0.0625 + 0.5625) = 0.3125.

Question 3

Use a &#64257;nite sum to estimate the average value of the function on the given interval by partitioning the interval and&#10;evaluating the function at the midpoints of the subintervals.&#10;

Accepted Answer

D

Question 4

Use a &#64257;nite sum to estimate the average value of the function on the given interval by partitioning the interval and&#10;evaluating the function at the midpoints of the subintervals.&#10;

Accepted Answer

To estimate the average value of $f(t) = 8 - \left( \cos \frac{\pi t}{2} \right)^2$ on $[0,4]$ with 4 subintervals, we divide the interval into 4 equal parts: $[0,1], [1,2], [2,3], [3,4]$. The midpoints are $0.5, 1.5, 2.5, 3.5$. Evaluating $f(t)$ at these midpoints and averaging gives us the estimate. Calculating $f(0.5), f(1.5), f(2.5), f(3.5)$ and averaging them gives $\frac{15}{2}$.

Question 5

Use a &#64257;nite sum to estimate the average value of the function on the given interval by partitioning the interval and&#10;evaluating the function at the midpoints of the subintervals.&#10;

Accepted Answer

The average value of a function $f(x)$ over an interval $[a, b]$ can be estimated by dividing the interval into $n$ subintervals, finding the midpoint of each subinterval, evaluating the function at these midpoints, and then averaging these values. Here, the interval $\left[\frac{1}{2}, \frac{11}{2}\right]$ is divided into 5 subintervals. The length of each subinterval is $\frac{\frac{11}{2} - \frac{1}{2}}{5} = 1$. The midpoints for these subintervals are $1, 2, 3, 4,$ and $5$. Evaluating $f(x) = \frac{5}{x}$ at these midpoints and averaging the results gives $\frac{1}{5} \left( \frac{5}{1} + \frac{5}{2} + \frac{5}{3} + \frac{5}{4} + \frac{5}{5} \right) = \frac{137}{60}$.

Question 6

Use a &#64257;nite sum to estimate the average value of the function on the given interval by partitioning the interval and&#10;evaluating the function at the midpoints of the subintervals.&#10;

Accepted Answer

We will use the midpoint rule to estimate the average value of the function. Dividing the interval into 5 subintervals gives us:

$$-3, \quad -3+2(2), \quad -3+2(4), \quad -3+2(6), \quad 7$$

or

$$-3, \quad -1, \quad 1, \quad 3, \quad 7$$

The width of each subinterval is 2. The midpoint of each subinterval is found by averaging the endpoints:

$$\frac{-3+(-1)}{2}=-2, \quad \frac{-1+1}{2}=0, \quad \frac{1+3}{2}=2, \quad \frac{3+7}{2}=5$$

Now we can evaluate the function at each midpoint:

$$f(-2)=(-2)^2-2=2$$
$$f(0)=0^2-2=-2$$
$$f(2)=2^2-2=2$$
$$f(5)=5^2-2=23$$

So the average value is approximately:

$$\frac{1}{5}\left(2-2+2+23 \right)=\frac{25}{5}=5$$

Therefore, the best choice is C) 10 since 5 is halfway between -3 and 7, the endpoints of the interval.

Question 7

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Accepted Answer

The answer of 11eb8c85_25d1_d6ee_8371_e9a641b63108_TB8583_00...

Question 8

&#10;A) 174&#10;B) 165&#10;C) 149&#10;D) 126

Accepted Answer

The interval from 4 to 8 is divided into 4 equal parts, each with a width of 1 (8 - 4 = 4, 4/4 = 1). For an upper sum, we use the right endpoint of each interval to calculate the area of each rectangle. The rectangles will be based at x = 5, 6, 7, and 8. The area of each rectangle is the function value at these points times the width (1). So, the total area is $(5^2 + 6^2 + 7^2 + 8^2) * 1 = (25 + 36 + 49 + 64) = 174$.

Question 9

&#10;A) 174&#10;B) 126&#10;C) 149&#10;D) 165

Accepted Answer

The interval from $x = 4$ to $x = 8$ is divided into 4 equal parts, each with a width of $1$ unit ($\frac{8 - 4}{4} = 1$). For a lower sum, we use the left end of each interval to calculate the area of each rectangle. The x-values at these points are 4, 5, 6, and 7. The area of each rectangle is the function value at these points times the width (1 unit). So, the total area is $$f(4)\times1 + f(5)\times1 + f(6)\times1 + f(7)\times1 = 4^2 + 5^2 + 6^2 + 7^2 = 16 + 25 + 36 + 49 = 126.$$

Question 10

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Accepted Answer

The answer of 11eb8c85_25d1_d6ed_8371_570e57a7f340_TB8583_00...

Question 11

Use a &#64257;nite sum to estimate the average value of the function on the given interval by partitioning the interval and&#10;evaluating the function at the midpoints of the subintervals.&#10;

Accepted Answer

The answer of Use a &#64257;nite sum to estimate the...

Question 12

Use a &#64257;nite sum to estimate the average value of the function on the given interval by partitioning the interval and&#10;evaluating the function at the midpoints of the subintervals.&#10;

Accepted Answer

The answer of Use a &#64257;nite sum to estimate the...

Question 13

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Accepted Answer

The answer of 11eb8c85_25d1_afdc_8371_ad7dc26e40a9_TB8583_00...

Question 14

f(x) = 1 - x2 between x = -1 and x = 1 using the &#34;midpoint rule&#34; with two rectangles of equal width.&#10;A) 0.75&#10;B) 0.5&#10;C) 2&#10;D) 1.5

Accepted Answer

The answer of f(x) = 1 - x2 between x...

Question 15

Use a &#64257;nite sum to estimate the average value of the function on the given interval by partitioning the interval and&#10;evaluating the function at the midpoints of the subintervals.&#10;

Accepted Answer

The answer of Use a &#64257;nite sum to estimate the...

Question 16

&#10;A) 86&#10;B) 54&#10;C) 62&#10;D) 69

Accepted Answer

The answer of  &#10;A) 86&#10;B) 54&#10;C) 62&#10;D) 69...

Question 17

&#10;A) 5&#10;B) 4.5&#10;C) 1&#10;D) 2.5

Accepted Answer

The answer of  &#10;A) 5&#10;B) 4.5&#10;C) 1&#10;D) 2.5...

Question 18

Use a &#64257;nite sum to estimate the average value of the function on the given interval by partitioning the interval and&#10;evaluating the function at the midpoints of the subintervals.&#10;

Accepted Answer

The answer of Use a &#64257;nite sum to estimate the...

Question 19

&#10;A) )75&#10;B) )625&#10;C) )125&#10;D) )3145

Accepted Answer

The answer of  &#10;A) )75&#10;B) )625&#10;C) )125&#10;D) )3145...

Question 20

Use a &#64257;nite sum to estimate the average value of the function on the given interval by partitioning the interval and&#10;evaluating the function at the midpoints of the subintervals.&#10;

Accepted Answer

The answer of Use a &#64257;nite sum to estimate the...

Question 21

Write the sum without sigma notation and evaluate it.&#10;

Accepted Answer

The answer of Write the sum without sigma notation and...

Question 22

Write the sum without sigma notation and evaluate it.&#10;

Accepted Answer

The answer of Write the sum without sigma notation and...

Question 23

A swimming pool has a leak. The leak is getting worse. The following table gives the leak rate every 6 hours. Suppose it keeps leaking 6.3 gallons every 6 hours. After losing 2700 gallons the leak is ﬁxed.
Approximately how long did the leak last? Use the right endpoints to estimate the ﬁrst 48
Hours.

A) 54.59 hours
B) 32.48 hours
C) 26.48 hours
D) -87.60 hours

Accepted Answer

The answer of A swimming pool has a leak. The...

Question 24

Write the sum without sigma notation and evaluate it.&#10;

Accepted Answer

The answer of Write the sum without sigma notation and...

Question 25

Write the sum without sigma notation and evaluate it.&#10;

Accepted Answer

The answer of Write the sum without sigma notation and...

Question 26

Write the sum without sigma notation and evaluate it.&#10;

Accepted Answer

The answer of Write the sum without sigma notation and...

Question 27

Estimate the value of the quantity.&#10;The table gives dye concentrations for a cardiac-output determination. The amount of dye injected was 4.6 mg. Plot the data and connect the data points with a smooth curve. Find the&#10;Area under the curve using rectangles. Use this area to estimate the cardiac output.  &#10;A) 285.7 L/min&#10;B) 12.6 L/min&#10;C) 0.1 L/min&#10;D) 6.3 L/min

Accepted Answer

The answer of Estimate the value of the quantity.&#10;The table...

Question 28

Joe wants to &#64257;nd out how far it is across the lake. His boat has a speedometer but no odometer. The table shows the boats velocity at 10 second intervals. Estimate the distance across the lake&#10;Using right-end point values.  &#10;A) 5900 ft&#10;B) 391 ft&#10;C) 3910 ft&#10;D) 4010 ft

Accepted Answer

The answer of Joe wants to &#64257;nd out how far...

Question 29

Write the sum without sigma notation and evaluate it.&#10;

Accepted Answer

The answer of Write the sum without sigma notation and...

Question 30

Estimate the value of the quantity.&#10;The table gives dye concentrations for a cardiac-output determination. The amount of dye injected was 5.0 mg. Plot the data and connect the data points with a smooth curve. Find the&#10;Area under the curve using rectangles. Use this area to estimate the cardiac output.  &#10;A) 381.6 L/min&#10;B) 18.9 L/min&#10;C) 9.4 L/min&#10;D) 0.2 L/min

Accepted Answer

The answer of Estimate the value of the quantity.&#10;The table...

Question 31

Estimate the value of the quantity.&#10;The table shows the velocity of a remote controlled race car moving along a dirt path for 8 seconds. Estimate the distance traveled by the car using 8 subintervals of length 1 with&#10;Right-end point values.  &#10;A) 172 in.&#10;B) 182 in.&#10;C) 188 in.&#10;D) 178 in.

Accepted Answer

The answer of Estimate the value of the quantity.&#10;The table...

Question 32

A hat is dropped from a hot air balloon. The hat falls faster and faster but its acceleration decreases over time due to air resistance. The acceleration is measured every second after the
Drop for 6 seconds. Find an upper estimate for the speed when t = 8 seconds.

A) 65.35 ft/sec
B) 33.98 ft/sec
C) 32.04 ft/sec
D) 192 ft/sec

Accepted Answer

The answer of A hat is dropped from a hot...

Question 33

A swimming pool has a leak. The leak is getting worse. The following table gives the leak rate every 6 hours. Suppose it keeps leaking 6.3 gallons every 6 hours. After losing 2700 gallons the leak is ﬁxed.
Approximately how long did the leak last? Use the right endpoints to estimate the ﬁrst 48
Hours.

A) 54.59 hours
B) 32.48 hours
C) 26.48 hours
D) -87.60 hours

Accepted Answer

The answer of A swimming pool has a leak. The...

Question 34

Write the sum without sigma notation and evaluate it.&#10;

Accepted Answer

The answer of Write the sum without sigma notation and...

Question 35

The velocity of a projectile &#64257;red straight into the air is given every half second. Use right endpoints to estimate the distance the projectile travelled in four seconds.  &#10;A) 471.8 m&#10;B) 982.8 m&#10;C) 491.4 m&#10;D) 943.6 m

Accepted Answer

The answer of The velocity of a projectile &#64257;red straight...

Question 36

A piece of tissue paper is picked up in gusty wind. The table shows the velocity of the paper at 2 second intervals. Estimate the distance the paper travelled using left-endpoints.  &#10;A) 256 ft&#10;B) 233 ft&#10;C) 236 ft&#10;D) 128 ft

Accepted Answer

The answer of A piece of tissue paper is picked...

Question 37

Write the sum without sigma notation and evaluate it.&#10;

Accepted Answer

The answer of Write the sum without sigma notation and...

Question 38

Estimate the value of the quantity.&#10;The table shows the velocity of a remote controlled race car moving along a dirt path for 8 seconds. Estimate the distance traveled by the car using 8 subintervals of length 1 with&#10;Left-end point values.  &#10;A) 328 in.&#10;B) 164 in.&#10;C) 169 in.&#10;D) 154 in.

Accepted Answer

The answer of Estimate the value of the quantity.&#10;The table...

Question 39

Write the sum without sigma notation and evaluate it.&#10;

Accepted Answer

The answer of Write the sum without sigma notation and...

Question 40

Write the sum without sigma notation and evaluate it.&#10;

Accepted Answer

The answer of Write the sum without sigma notation and...

Question 41

Provide an appropriate response.&#10;

Accepted Answer

The answer of Provide an appropriate response.&#10; ...

Question 42

Express the sum in sigma notation.&#10;

Accepted Answer

The answer of Express the sum in sigma notation.&#10; ...

Question 43

Find the value of the speci&#64257;ed &#64257;nite sum.&#10;

Accepted Answer

The answer of Find the value of the speci&#64257;ed &#64257;nite...

Question 44

Evaluate the sum.&#10;

Accepted Answer

The answer of Evaluate the sum.&#10; ...

Question 45

Provide an appropriate response.&#10;

Accepted Answer

The answer of Provide an appropriate response.&#10; ...

Question 46

Express the sum in sigma notation.&#10;

Accepted Answer

The answer of Express the sum in sigma notation.&#10; ...

Question 47

Find the value of the speci&#64257;ed &#64257;nite sum.&#10;

Accepted Answer

The answer of Find the value of the speci&#64257;ed &#64257;nite...

Question 48

Evaluate the sum.&#10;

Accepted Answer

The answer of Evaluate the sum.&#10; ...

Question 49

Evaluate the sum.&#10;

Accepted Answer

The answer of Evaluate the sum.&#10; ...

Question 50

Find the value of the speci&#64257;ed &#64257;nite sum.&#10;

Accepted Answer

The answer of Find the value of the speci&#64257;ed &#64257;nite...

Question 51

Evaluate the sum.&#10;

Accepted Answer

The answer of Evaluate the sum.&#10; ...

Question 52

Express the sum in sigma notation.&#10;

Accepted Answer

The answer of Express the sum in sigma notation.&#10; ...

Question 53

Evaluate the sum.&#10;

Accepted Answer

The answer of Evaluate the sum.&#10;   ...

Question 54

Evaluate the sum.&#10;

Accepted Answer

The answer of Evaluate the sum.&#10; ...

Question 55

Provide an appropriate response.&#10;

Accepted Answer

The answer of Provide an appropriate response.&#10; ...

Question 56

Find the value of the speci&#64257;ed &#64257;nite sum.&#10;

Accepted Answer

The answer of Find the value of the speci&#64257;ed &#64257;nite...

Question 57

Express the sum in sigma notation.&#10;

Accepted Answer

The answer of Express the sum in sigma notation.&#10; ...

Question 58

Express the sum in sigma notation.&#10;

Accepted Answer

The answer of Express the sum in sigma notation.&#10; ...

Question 59

Provide an appropriate response.&#10;

Accepted Answer

The answer of Provide an appropriate response.&#10; ...

Question 60

Find the value of the speci&#64257;ed &#64257;nite sum.&#10;

Accepted Answer

The answer of Find the value of the speci&#64257;ed &#64257;nite...

Question 61

Evaluate the sum.&#10;

Accepted Answer

The answer of Evaluate the sum.&#10;   ...

Question 62

Express the limit as a de&#64257;nite integral.&#10;

Accepted Answer

The answer of Express the limit as a de&#64257;nite integral.&#10;...

Question 63

Find the formula and limit as requested.&#10;

Accepted Answer

The answer of Find the formula and limit as requested.&#10;...

Question 64

Evaluate the sum.&#10;

Accepted Answer

The answer of Evaluate the sum.&#10;   ...

Deck 5: Integration