Deck 5: Key Concept- the Definite Integral

Question 1

Estimate $\int_{3}^{4} \ln x d x$ to 1 decimal place, choosing a suitable $\Delta$ x.

Answer 1

1.2

Question 2

At time t, in seconds, the velocity, v, in miles per hour, of a car is given by $v(t)$ for $0 \leq t \leq 8$ .A second car travels exactly 15 miles per hour faster than the first car.Using $\Delta$ t = 2, the right-hand estimate for the distance traveled during this time by the first car is 143 miles.What is the right-hand estimate for the distance traveled during this time by the second car?

Answer 2

263 miles

Question 3

Consider the region A shown in the following graph.Is the area of A more or less than 0.7?

Answer 3

less

Question 4

At time t, in seconds, your velocity v, in meters/sec, is given by $v(t)=9+4 t^{2}$ for $0 \leq t \leq 6$ .Which is more accurate?

A)An estimate of the distance traveled during this time using $\Delta$ t = 1.
B)An estimate of the distance traveled during this time using F $\Delta$ t = 2.

Question 5

The figure below shows the graph of the velocity, v, of an object (in meters/sec.).If the graph were shifted up two units, what would it mean for the motion of the object?

A)The velocity at each time would be 2 m/sec greater.
B)The velocity at each time would be 12 m/sec greater.
C)The velocity at each time would be 20 m/sec greater.
D)The velocity at each time would be 120 m/sec greater.
E)The velocity at each time would be the same.

Question 6

What does the following figure represent?

A)The right-hand Riemann sum for the function f on the interval 0 $\le$ t $\le$ 12 with $\Delta$ t = 3.
B)The right-hand Riemann sum for the function f on the interval 0 $\le$ t $\le$ 12 with $\Delta$ t = 6.
C)The left-hand Riemann sum for the function f on the interval 0 $\le$ t $\le$ 12 with $\Delta$ t = 3.
D)The left-hand Riemann sum for the function f on the interval 0 $\le$ t $\le$ 12 with $\Delta$ t = 6.

Question 7

Consider a sports car which accelerates from 0 ft/sec to 88 ft/sec in 5 seconds (88 ft/sec = 60mph).The car's velocity is given in the table below.
Find the lower bound for the distance the car travels in 5 seconds.

Question 8

Two greyhound racing dogs, A and B, start at the same time and travel in the same direction along a straight track.The figure below gives the velocity, v, of each dog as a function of time t.Which dog travels the farthest?

Question 9

At time t, in seconds, your velocity v, in meters/sec, is given by $v(t)=4+4 t^{2}$ for $0 \leq t \leq 6$ .Use $\Delta$ t = 1 to estimate distance during this time.(Average right- and left-hand sums).

Question 10

Estimate the area of the region above the curve $y=7 \cos x$ and below y = 7 for 0 $\le$ x $\le$ $\pi$ /2.Round to 2 decimal places.

Question 11

The figure below shows the graph of the velocity, v, of an object (in meters/sec.).If the graph were shifted up 4 units, how would the total distance traveled between t = 0 and t = 6 change?

A)It would increase by 4 units.
B)It would increase by 24 units.
C)It would remain the same.
D)It would decrease by 4 units.
E)It would decrease by 24 units.

Question 12

Use the table to estimate $\int_{0}^{50} f(x) d x$ with n = 5 and $\Delta$ x = 10.(Average left-and right-hand sums).
$\begin{array}{ccccccc}x & 0 & 10 & 20 & 30 & 40 & 50 \\f(x) & 40 & 45 & 55 & 60 & 80 & 95\end{array}$

Question 13

The graph shown below is that of the velocity of an object (in meters/second).Find a lower estimate of the total distance traveled from t = 0 to t =5 seconds .

Question 14

A car is observed to have the following velocities at times t = 0, 2, 4, 6:
Give the lower estimate for the distance the car traveled.

Question 15

If an upper estimate of the area of a region bounded by the curve in the following figure, the horizontal axis and the vertical lines x = 3 and x = -3 is 15, what is the upper estimate if the graph is shifted up one unit?

Question 16

At time t, in seconds, your velocity v, in meters/sec, is given by $v(t)=2+5 t^{2}$ for $0 \leq t \leq 6$ .Use $\Delta$ t = 2 to estimate distance during this time.(Average right- and left-hand sums).

Question 17

Consider a sports car which accelerates from 0 ft/sec to 88 ft/sec in 5 seconds (88 ft/sec = 60mph).The car's velocity is given in the table below. $\begin{array}{cccccc}t & 0 & 1 & 2 & 3 & 4 &5\\V(t) & 0 & 30 & 52 & 68 & 80&88\end{array}$ In which time interval is the average acceleration smallest?

A)Between t = 4 and t = 5.
B)Between t = 3 and t = 4.
C)Between t = 2 and t = 3.
D)Between t = 1 and t = 2.
E)Between t = 0 and t = 1.

Question 18

Using the following figure, calculate the value of the right-hand Riemann sum f on the interval 0 $\le$ t $\le$ 12 with $\Delta$ t = 6.

Question 19

A particle starting at the origin moves along the x-axis such that its velocity along the line can be modeled by the function v(t)= 2 - 6t units/sec.What is the exact change in position by the function from t = 0 to t = 1.333333?

A)-2.667 units
B)-5.333 units
C)-4.333 units
D)-6.333 units
E)-6.833 units

Question 20

At time t, in seconds, the velocity, v, in miles per hour, of a car is given by $v(t)=8+0 .8 t^{2}$ for $0 \leq t \leq 8$ .Use $\Delta$ t = 2 to estimate the distance traveled during this time.Give the right-hand sum.

Question 21

What is the value of $\int_{a}^{c} f(x) d x$ if the area of A = 15 and the area of B = -4?

A)5.5
B)22
C)-11
D)11
E)19

Question 22

What is the value of $\int_{a}^{c}|f(x)| d x$ if the area of A = 9 and the area of B = -2?

A)3.5
B)14
C)-7
D)7
E)11

Question 23

Below is the graph of the rate r in arrivals/minute at which students line up for breakfast at the Cafeteria Charlotte.The first people arrive at 6:50a.m.and the line opens at 7:00a.m.Suppose that once the line is open, checkers can check peoples' meal cards at a constant rate of 20 people per minute.Use the graph and this information to find an estimate for the length of the line (i.e.the number of people)at 7:10.

A)200
B)230
C)260
D)290

Question 24

Estimate the area of the region between y = cos x, y = 3x, x = - $\pi$ /2, and x = 0.Round to 3 decimal places.

Question 25

Estimate the area of the region under the curve and above the x-axis for .Round to 3 decimal places.

Question 26

Which of the following best approximates $\frac{5}{\pi} \int_{0}^{\pi / 2} \sin t d t$ ?

A)1.44
B)1.49
C)1.54
D)1.59
E)1.64

Question 27

A car is moving along a straight road from A to B, starting from A at time t = 0.Below is the velocity (positive direction is from A to
B)plotted against time.

How many kilometers away from A is the car at time t = 9?

Question 28

A shop is open from 9am-7pm.The function r(t), graphed below, gives the rate at which customers arrive (in people/hour)at time t.Suppose that the salespeople can serve customers at a rate of 60 people per hour.When do people have to start waiting in line before being served? Answer to the nearest half-hour.You do not need to include "am" or "pm".

Question 29

Calculate the area under the curve y= 3x +3 for values between [1, 4].

A)15.75
B)157.5
C)993.25
D)10.5
E)31.5

Question 30

Find the average value of over [1, 3].

Question 31

The average value of a function g on 0 $\le$ x $\le$ 2 is a constant $\bar{g}$ given by $\bar{g}=\frac{1}{2-0} \int_{0}^{2} g(d) d x$ .Also, $\int_{0}^{2}(g(x)-\bar{g})^{2} d x \geq 0$ , since $(g(x)-\bar{g})^{2}$ is a square.Which of the following must be true?

A) $\left(\int_{0}^{2} g(x) d x\right)^{2} \leq \int_{0}^{2}(g(x))^{2} d x$
B) $\left(\int_{0}^{2} g(x) d x\right)^{2} \geq \int_{0}^{2}(g(x))^{2} d x$
C) $\left(\int_{0}^{2} g(x) d x\right)^{2}=\int_{0}^{2}(g(x))^{2} d x$
D)None of these

Question 32

Below is the graph of the velocity, in feet per second, of a hat that is thrown up in the air from ground level.Positive velocity means upward motion. [Note that this is the graph of velocity, not distance.] About how big is the average speed over the first 4 seconds?

A)13.1 ft/sec
B)10.8 ft/sec
C)6.8 ft/sec
D)4.4 ft/sec

Question 33

Does the quantity $\int_{a}^{a+h} f(x) d x$ represent a length or an area in the picture?

A)An area
B)A length

Question 34

A car is moving along a straight road from A to B, starting from A at time t = 0.Below are graphs of the velocity and the acceleration plotted against time (positive direction is from A to B)

A)The one on the right
B)The one on the left. Which graph shows the velocity?

Question 35

Find the average value of over [0, 2].Round to the nearest whole number.

Question 36

The velocity of an object is given by $v(t)$ , and the acceleration is given by $a(t)$ .What is the relationship between the total change in v(t)over the interval 0 $\le$ t $\le$ 10 and a(t)?

A) $\int_{0}^{10} v(t) d t=a(10)-a(0)$
B) $\int_{0}^{10} a(t) d t=v(10)-v(0)$
C) $\int_{0}^{10} t d t=\alpha(10)-v(10)$
D) $\int_{0}^{10} v(t) d t=\int_{0}^{10} a(t) d t$

Question 37

How can the quantity f(a)h be represented in the picture?

A)By the slope of TV
B)By the length of UR
C)By the area of PQRS
D)By the area of QRU

Question 38

Rashmi and Tia both go running from 7:00am to 8:00am.Both women increase their velocity throughout the hour, both beginning at a rate of 9 mi/hr.at 7:00am and running at a rate of 13 mi/hr by 8:00am.Tia's velocity increases at an increasing rate and Rashmi's velocity increases at a decreasing rate.Who has the greatest average velocity? If they had the same average velocity, enter "same".

Question 39

The velocity and acceleration of an object are given by the graphs shown below, where $v(0)=0$ .Which graph shows acceleration?

A)The one on the left.
B)The one on the right.

Question 40

The average value of a function g on 0 F $\le$ x $\le$ 3 is a constant $\bar{g}$ given by $\bar{g}=\frac{1}{3-0} \int_{0}^{3} g(x) d x$ .Which of the following must be true?

A) $\int_{0}^{3} \bar{g} g(x)<(\bar{g})^{2}$
B) $\int_{0}^{3} \bar{g} g(x)>(\bar{g})^{2}$
C) $\int_{0}^{3} \bar{g} g(x)=(\bar{g})^{2}$
D)None of the above

Question 41

If and is odd, what is ?

Question 42

If r(t)represents the rate at which a country's debt is growing, then the increase in its debt between 1990 and 2000 is given by

A) $\frac{r(2000)-r(1990)}{2000-1990}$
B) $r(2000)-r(1990)$
C) $\frac{1}{10} \int_{1990}^{2000} r(t) d t$
D) $\int_{1990}^{2000} r(t) d t$
E) $\frac{1}{10} \int_{1990}^{2000} r^{\prime}(t) d t$

Question 43

A potato is cooking in an oven.Explain in words what means if is the temperature of the potato, in degrees Farenheit, and t is time, in minutes.

Question 44

Explain in words what means if is velocity in miles/hour and t is time, in hours.

Question 45

If $E(t)$ represents the energy consumed in a household in watts/month, what are the units of $\int_{a}^{3} E(t) d t$ ?

A)watts
B)watts/month
C)watts/month²
D)month
E)watts²/month²

Question 46

Suppose , , , and .Find .

Question 47

If $f(x)=\frac{x}{2}+4$ and g(x)= x + 1, how do $\int_{8}^{12} f(x) d x$ and $\int_{8}^{12} g(x) d x$ compare?

A) $\int_{8}^{12} f(x) d x$ < $\int_{8}^{12} g(x) d x$
B) $\int_{8}^{12} f(x) d x$ > $\int_{8}^{12} g(x) d x$
C) $\int_{8}^{12} f(x) d x$ = $\int_{8}^{12} g(x) d x$

Question 48

If and , what is ?

Question 49

If and is even, and ?what is ?

Question 50

Suppose A = the area under the curve $y=e^{-x^{2}}$ over the interval -5 $\le$ x $\le$ 5.Which of the following is/are true?

A) $A=F(5)-F(-5)$ , where $F(x)=-\frac{e^{-x^{2}}}{2 x}$ .
B) $1.773$$<$$A$
C) $1.771<$$A<1.773$
D) $1.769<$$A<1.771$

Question 51

If and , evaluate .

Question 52

Evaluate the definite integral .

Question 53

If f is even and , what is ?

Question 54

The average value of y = h(x)equals a for 0 $\le$ x $\le$ 5, and equals b for 5 $\le$ x $\le$ 15.What is the average value of h(x)for 0 $\le$ x $\le$ 15?

A) $\frac{a+b}{2}$
B) $\frac{a+2 b}{3}$
C) $\frac{a+b}{15}$
D) $\frac{a+2 b}{15}$

Question 55

What is the value of ?

Question 56

Evaluate

Question 57

Let $\int_{0}^{8} f(x) d x=C$ .If f(x)is even, what is $\int_{-8}^{8} f(x) d x$ ?

A) $C$
B) $8 C$
C) $16 C$
D) $2 C$
E)0

Question 58

Suppose f(t)is given by the graph below.If , what is ?

Question 59

Suppose , , , and .Find .

Question 60

Let $\int_{0}^{10} f(x) d x=C$ .What is the average value of f(x)on the interval x = 0 to x = 10?

A) $10 C$
B)C
C) $\frac{C}{10}$
D)0

Question 61

If a function is concave up, then the left-hand Riemann sums are always less than the right-hand Riemann sums with the same subdivisions, over the same interval.

Question 62

If $\int_{a}^{b} f(x)=0$ , then f must have at least one zero between a and b (assume a $\neq$ b).