Deck 6: Applications of the Definite Integral

Question 1

Identify the graph and area of the region bounded by the curves . Use only one integral. The figures below are not necessarily to scale.

A)
B)
C)
D)

Answer 1

Question 2

A 2-person tent has a rectangular base and rises to a single point 5' up. Compute the volume of the tent. 7 ft 8 ft

A) 75 ft³
B) 70 ft³
C) 140 ft³
D) 280 ft³

Answer 2

140 ft³

Question 3

Identify the graph and the area bounded by the curves

A) area = 5/3
B) area = 1/6
C) area = 1/3
D) area = 1/3

Answer 3

area = 1/3

Question 4

Find the area between the curves on the given interval. y = x⁴, y = x - 1, -1 $\leq$ x $\leq$ 1

A)
B)
C)
D)

Question 5

Identify the graph and area of the region bounded by the curves . The figures below are not necessarily to scale.

A) area =
B) area =
C) area =
D) area =

Question 6

Estimate the intersection points to the nearest tenth then use these points to estimate the area.

A) area
B) area
C) area
D) area

Question 7

Find the volume of the solid with cross-sectional area extending over the range .

A)
B)
C)
D)

Question 8

Compute the volume of the solid formed by revolving the region bounded by and about the x-axis.

A)
B)
C)
D)

Question 9

Identify the graph and the area bounded by the curves on the interval .

A)
B)
C)
D)

Question 10

Find the volume of the solid with cross-sectional area extending over the range .

A)
B)
C)
D)

Question 11

Find the area between the following curves on the given interval.

A)
B)
C)
D)

Question 12

Identify the graph and the area of the region determined by the intersections of the tabulated curves using Simpson's Rule. x
0)00
0)25
0)50
0)75
1)00
F₁(x)
1)0
0)25
0)07
0)02
0)00
F₂(x)
1)0
0)75
0)50
0)25
0)00

A)
B)
C)
D)

Question 13

Identify the graph and area of the region bounded by the curves . Use only one integral. The figures below are not necessarily to scale.

A)
B)
C)
D)

Question 14

Find the area of the region determined by the intersections of the curves.

A) 10
B) 20
C)
D)

Question 15

Find the area between the following curves on the given interval.

A)
B)
C)
D)

Question 16

Find the area of the region bounded by the given curves. Write a single integral that represents the area.

A) area
B) area
C) area
D) area

Question 17

Use Simpson's rule to estimate the volume from the given cross sectional areas. x(cm)
-1)0
-0)5
0)0
0)5
1)0
A(x) (cm²)
1)4
1)5
2)0
2)5
2)6

A) 3.3
B) 4.0
C) 12.0
D) 6.0

Question 18

A football (American) has a circular cross section with radius approximately for . Compute the volume of the football.

A) 238.9 inches³
B) 165.9 inches³
C) 119.4 inches³
D) 82.9 inches³

Question 19

Compute the volume of a solid formed by revolving the region bounded by

A)
B)
C)
D)

Question 20

Given a certain amount of land potentially devoted to farming, one will first use the more fertile regions where production costs are lowest before using the less fertile regions. The consequence of this logical strategy is that the production costs increase as production increases. A certain landowner has enough land to produce 10⁶ bushels of a particular crop, and the production costs, c, of the n^th bushel is described with the equation . If the market price is p for this particular crop, the profit for producing the n^th bushel is . If p = $5.25, how much total profit could the farmer expect to obtain by producing bushels.

A) $377,600
B) $289,200
C) $350,000
D) $322,400

Question 21

Use the best method available to find the volume of the solid formed by revolving the region bounded by and about the x-axis. Write the integral that is used to find the volume.

Question 22

Use cylindrical shells to compute the volume of the solid formed by revolving the region bounded by about x = 5.

A)
B)
C)
D)

Question 23

Use the best method available to find the volume of the solid formed by revolving the region bounded by the y-axis, and about the y-axis. Write the integral that is used to find the volume.

Question 24

Show that the volume of a solid formed by revolving the region bounded by x = y² and x = r about x = r is .

Question 25

Use cylindrical shells to compute the volume of the solid formed by revolving the region bounded by about y = 10.

A)
B)
C)
D)

Question 26

Use Simpson's Rule to estimate the volume of the shape obtained by revolving the cross-section given in the table about the y-axis. y
0
0)4375
0)75
0)9375
1
0)9375
0)75
0)4375
0
X
1
1)25
1)5
1)75
2
2)25
2)5
2)75
3

A) 3
B) 17
C) 24
D) 26

Question 27

Find the volume of the solid formed by revolving the region bounded by about y = 0.

A)
B)
C)
D)

Question 28

Let R be the region bounded by , the x-axis, and the y-axis. Compute the volume of the solid formed by revolving R about the y-axis.

A)
B)
C)
D)

Question 29

Let R be the region bounded by , the x-axis, and the y-axis. Compute the volume of the solid formed by revolving R about the x-axis.

A)
B)
C)
D)

Question 30

The base of a solid V is the region bounded by and Find the volume if V has square cross sections.

A)
B)
C)
D)

Question 31

Find the volume of the solid formed by revolving the region bounded by about x = 8.

A)
B)
C)
D)

Question 32

Use the best method available to find the volume of the solid formed by revolving the region bounded by and about the x-axis. Write the integral that is used to find the volume.

A)
B)
C)
D)

Question 33

The base of a solid V is the region bounded by and Find the volume if V has semicircular cross sections perpendicular to the x-axis.

A)
B)
C)
D)

Question 34

Let R be the region bounded by . Compute the volume of the solid formed by revolving R about x = 2.

A)
B)
C)
D)

Question 35

Sketch the solid bounded by and the x-axis on the interval revolved about the line Draw a typical shell and write an integral that can be used to compute the volume of the solid.

Question 36

Use cylindrical shells to compute the volume of the solid formed by revolving the region bounded by about x = 7.

A)
B)
C)
D)

Question 37

A solid is formed by revolving the given region about the given line. Round to three decimal places. Region bounded by y = sec x, y = 0, x = , and x = about y = -6

A) 72.737
B) 23.153
C) 250.39
D) 19.348

Question 38

Let R be the region bounded by . Compute the volume of the solid formed by revolving R about y = 20.

A)
B)
C)
D)

Question 39

Use the best method available to find the volume of the solid formed by revolving the region bounded by and about the y-axis. Write the integral that is used to find the volume.

Question 40

Find the volume of the solid formed by revolving the region bounded by about x = 5.

A)
B)
C)
D)

Question 41

Set up the integral for the surface area of the surface of revolution and approximate the integral with a numerical method. Round answers to two decimal places. revolved about the x-axis

A)
B)
C)
D)

Question 42

Identify the integral used to determine the surface area of the surface of revolution for the shape described by , , revolved about the x-axis.

A)
B)
C)
D)

Question 43

A person 6' tall wants to buy a jump rope. If , when the rope is at its lowest point, his hands will be 2' apart and 3' above the ground, and if the rope will take on the shape of a parabola just barely hitting the ground, how long must the rope be? [Consider the form of the rope to be described by the equation , with the origin being a spot on the ground underneath the jumper.]

A) 6.75 feet
B) 6.50 feet
C) 6.25 feet
D) 6.00 feet

Question 44

In a movie stunt, a car is driven off a cliff and falls into the ocean 200 feet below. If the car is going 50 feet/second (horizontally) when it goes over the cliff, its horizontal position is described as x = 50 t, where t is the time in seconds. Its vertical position is described as y = 16 t², where y is the distance below the cliff (i.e. y = 200 when the car hits the ocean). Compute the length of the path traversed by the falling car. Round to the nearest foot. [Hint: The two equations will need to be combined, eliminating t so that y is expressed as a function of x.]

A) 274 feet
B) 279 feet
C) 284 feet
D) 289 feet

Question 45

Identify the region (solid lines) and axis of revolution (dashed line) for the solid whose volume is described by the integral .

A)
B)
C)
D)

Question 46

Two divers, one on the 15 ft platform and one on the 30 ft platform, hope to perform a joint dive in which they enter the water at the same time. If each begins with an upward jump with an initial velocity of at 14 ft/s, how much time difference should there be between the start of the two dives?

A) 0.43 s
B) 30.00 s
C) 0.38 s
D) 0.75 s

Question 47

A solid shape described as the region bounded by revolved around the y-axis, needs to have 1/4 of its volume trimmed off to meet a weight restriction. This will be done by reducing the radius of the piece - in other words, the resulting shape will be described as above, but with the region also bounded by x = q. Find the resulting radius, q.

A)
B)
C)
D)

Question 48

Compute the arc length exactly.

A)
B)
C)
D)

Question 49

Identify the initial conditions y(0) and y'(0) for the vertical motion of an object , if the object thrown downward at a velocity of 6 feet per second from a height of 30 feet. [Take the origin to be on the ground.]

A) y(0) = 30, y'(0) = -6
B) y(0) = 0, y'(0) = -6
C) y(0) = 30, y'(0) = 6
D) y(0) = 0, y'(0) = 6

Question 50

Set up the integral for arc length and then approximate the integral with a numerical method. Round answers to four decimal places.

Question 51

A 0.5 kg rock is dropped from the edge of a 52-foot cliff. How fast will it be going when it hits the ground at the base of the cliff?

A) ft/s
B) ft/s
C) 52 ft/s
D) ft/s

Question 52

Approximate the length of the curve using 4 secant lines. Round to three decimal places. y = x⁴ 0 $\leq$ x $\leq$ 2

A) 32.301
B) 7.060
C) 16.608
D) 14.119

Question 53

Pick the integral that represents a volume calculation by cylindrical shells that is equivalent to the following integral that represents a calculation of the same volume but by the method of washers.

A)
B)
C)
D)

Question 54

Compute the arc length exactly. , -2 $\leq$ x $\leq$ -1

A)
B)
C)
D)

Question 55

Identify the integral used to determine the surface area of the surface of revolution for the shape described by , , revolved about the x-axis.

A)
B)
C)
D)

Question 56

Molding is typically long, thin, decorative pieces of wood used, for instance, around windows or doors. The cross-section of a particular type of molding can be described as the region bounded by , x = 4, x = 0, and (all measured in inches), projected perpendicular to the xy-plane (see figure). If the manufacturer paints all four sides (but not the ends) and each gallon of paint covers 250 ft², how many linear feet of molding can be coated with one gallon?

A) 315.8 feet
B) 264.2 feet
C) 512.4 feet
D) 128.1 feet

Question 57

Identify the initial conditions y(0) and y'(0) for the vertical motion of an object, if the object is dropped from a height of 45 feet. [Take the origin to be on the ground.]

A) y(0) = 0, y'(0) = 0
B) y(0) = 45, y'(0) = 45t
C) y(0) = 0, y'(0) = -45
D) y(0) = 45, y'(0) = 0

Question 58

Find the length of the curve defined by function and extending from x = 0 to x = 4. Round your answer to three decimal places.

A) 12.228
B) 12.328
C) 12.428
D) 12.528

Question 59

An anthill is in the shape formed by revolving the region bounded by and the x-axis about the y-axis. A researcher removes a cylindrical core from the center of the hill. What should the radius be to give the researcher 5% of the dirt?

A) 0.8409
B) 0.6818
C) 0.1591
D) 0.3182

Question 60

Identify the region (solid lines) and axis of revolution (dashed line) for the solid whose volume is described by the integral .

A)
B)
C)
D)

Question 61

An object propelled from the ground with an initial velocity of 95 ft/s will reach a maximum height of 141.0 ft. If the initial velocity is increased 13%, by what percentage will the maximum height increase? Round percentages to the nearest integer.

A) 13%
B) 28%
C) 2%
D) 22%

Question 62

A crash test is performed on a vehicle. The force of the wall on the front bumper is shown in the table. Estimate the impulse and the speed of the vehicle to the nearest whole number (use m = 200 slugs).

A) impulse: 246,000 velocity: 1230 ft/s
B) impulse: 8200 velocity: 41 ft/s
C) impulse: 10,800 velocity: 54 ft/s
D) impulse: 324,000 velocity: 1620 ft/s

Question 63

A baseball pitcher releases the ball horizontally from a height of 5 ft with an initial speed of 120 ft/s. Find the height of the ball when it reaches home plate 60 feet away. Round to two decimal places. (Hint: Determine the time of flight from the x-equation, then use the y-equation to determine the height.)

A) 2.00 ft
B) 1.00 ft
C) 4.00 ft
D) 3.00 ft

Question 64

A force of 7 pounds stretches a spring 7 inches. Find the work done in stretching this spring 10 inches beyond its natural length.

A) foot-pounds
B) foot-pounds
C) foot-pounds
D) foot-pounds

Question 65

A projectile passes overhead, and you attempt to shoot it down with a cannon. When it passes overhead it is traveling horizontally at 100 m/s at an elevation of 1000 m. Your cannon launches shells at 200 m/s, and in this instance you fire just as the projectile passes overhead and at an angle of 60^o. In this scenario the horizontal velocities of the two projectiles are the same. At what elevation will the cannon shell and the other projectile collide?

A) They won't collide before the cannon shell hits the ground.
B) 971.7 m
C) 877.5 m
D) 836.7 m

Question 66

A certain not-so-wiley coyote discovers that he just stepped off the edge of a cliff. Fourteen seconds later he hits the ground in a puff of dust. How high in meters is the cliff?

A) 1920.8 m
B) 960.4 m
C) 68.6 m
D) 137.2 m

Question 67

Where is the center of mass of a region of uniform density bounded by ?

A)
B)
C)
D)

Question 68

When pumping water out of a full hemispherical basin with radius r feet, how far down will the water level be when 1/4 of the work has been done? [The density of water is 62.4 lbs/ft³.]

A)
B)
C)
D)

Question 69

A ball is thrown at an angle of with an initial speed of 25 meters/second. What is its time of flight ? [Assume the ball is thrown from ground level and lands at ground level.]

A) 0.14 seconds
B) 3.13 seconds
C) 1.10 seconds
D) 3.61 seconds

Question 70

An object is dropped from a height of 140 feet. Another object directly below the first is launched vertically from the ground with an initial velocity of 70 ft/s. Determine when and how high up the objects collide.

Question 71

Compute the weight in ounces of an object extending from x = -5 to x = 25 with density slugs/in. Round answer to nearest whole number.

A) 21 oz
B) 4 oz
C) 86 oz
D) 443 oz

Question 72

A water tower is spherical in shape with radius 50 feet, extending from 100 to 200 feet above ground. Compute the work done in filling the tank from the ground.

Question 73

A 30-foot chain weighs 750 pounds and is hauled up to the deck of a boat. The chain is oriented vertically and the top of the chain starts 30 feet below the deck. Compute the work done.

A) 11,250 foot-pounds
B) 22,875 foot-pounds
C) 375 foot-pounds
D) 33,750 foot-pounds

Question 74

In a baseball game a base-runner can attempt to advance when a fielder catches a hit ball before it reaches the ground, so long as the runner doesn't leave base before the fielder catches the ball. The fielder then attempts to throw the ball to the base ahead of the runner. If a base-runner can run from 3rd base to home plate in 4.6 sec, how fast must the centerfielder's throw leave his hand if he makes the catch then immediately throws the ball 370 ft from home plate? Assume the base-runner leaves his base at the same instant the centerfielder throws the ball, and that the ball reaches the catcher standing over home plate at the same height from which it was thrown and at the same time the runner would reach the plate. [Hint: This is basically a problem of determining how much initial vertical velocity the ball needs to stay in the air long enough to reach the plate.]

A) 80 ft/sec
B) 74 ft/sec
C) 109 ft/sec
D) 154 ft/sec

Question 75

When a certain spring is stretched 0.20 m, 7 of work is done. What is the spring's "spring constant"?

A) 350.0 N/m
B) 175.0 N/m
C) 35.0 N/m
D) 17.5 N/m

Question 76

A cylindrical water tank has a radius of 4 feet and a height of 3.0 feet. Compute the work done to pump the water out of a filled tank through the top. [The density of water is 62.4 lbs/ft³.]

A) 14,115 ft-lbs
B) 4705 ft-lbs
C) 281 ft-lbs
D) 7390 ft-lbs

Question 77

What is the minimum speed a ball must be thrown upward if it is to reach a height of 13 feet?

A) 13 ft/s
B) ft/s
C) ft/s
D) 0 ft/s

Question 78

A projectile traveling with momentum 4 kg m/s in one direction undergoes an impulse in the opposite direction. The force of the impulse is described as 6000(sin500t) kg m/s² and it is exerted over a time range . What is the object's momentum (in the direction of the impulse) after the impulse?

A) 8 kg m/s
B) -20 kg m/s
C) -19 kg m/s
D) 16 kg m/s

Question 79

Starting with the expression for work , one can change the look of the expression without changing what it represents by dividing the force F by the area it is applied over and multiplying the differential distance dx by that same area. Force divided by area is a pressure (P), and area times the differential distance is a differential volume. Hence one can also describe work as . A gas at constant temperature will change its pressure inversely to changes in volume, . If a sample of gas has a pressure of 1 atmosphere when its volume is 1 L, how much work does it do when it expands from 1 L to 3 L?

A)
B)
C)
D)

Question 80

Identify the mass of a thin wire with density .

A)
B)
C)
D)