Deck 7: Applications of Integration

Question 1

Solve the differential equation.

A)
B)
C)
D)
E)

Answer 1

Question 2

The masses are located at the point .Find the moments and and the center of mass of the system. ;

A)
B)
C)
D)
E)

Answer 2

Question 3

Calculate the center of mass of the lamina with density

Answer 3

(0,2)

Question 4

Find the coordinates of the centroid for the region bounded by the curves , x = 0, and y = 72.

Question 5

A tank contains 1050 L of brine with 20 kg of dissolved salt.Pure water enters the tank at a rate of 16 L/min.The solution is kept thoroughly mixed and drains from the tank at the same rate.How much salt is in the tank after 20 minutes?

Question 6

A swimming pool is 10 ft wide and 36 ft long and its bottom is an inclined plane,the shallow end having a depth of 1 ft and the deep end,12 ft.If the pool is full of water,find the hydrostatic force on the shallow end.(Use the fact that water weighs 62.5 lb/ .)

Question 7

Find the exact coordinates of the centroid.

Question 8

Choose the differential equation corresponding to this direction field.

A)
B)
C)
D)
E)

Question 9

Find the centroid of the region bounded by the graphs of the given equations.

A)
B)
C)
D)

Question 10

Find the centroid of the region bounded by the given curves.

A)
B)
C)
D)
E)

Question 11

Solve the initial-value problem.

Question 12

One model for the spread of an epidemic is that the rate of spread is jointly proportional to the number of infected people and the number of uninfected people.In an isolated town of 2000 inhabitants, 110 people have a disease at the beginning of the week and 1100 have it at the end of the week.How long does it take for 70% of the population to be infected?

A)
B)
C)
D)
E)

Question 13

The Pacific halibut fishery has been modeled by the differential equation where is the biomass (the total mass of the members of the population)in kilograms at time t (measured in years),the carrying capacity is estimated to be and per year.If ,find the biomass a year later.

Question 14

Find the centroid of the region bounded by the given curves.

A)
B)
C)
D)
E) None of these

Question 15

Find the solution of the differential equation that satisfies the initial condition .

Question 16

Solve the differential equation.

A)
B)
C)
D)
E) None of these

Question 17

A certain small country has $20 billion in paper currency in circulation,and each day $70 million comes into the country's banks.The government decides to introduce new currency by having the banks replace old bills with new ones whenever old currency comes into the banks.Let denote the amount of new currency in circulation at time t with .Formulate and solve a mathematical model in the form of an initial-value problem that represents the "flow" of the new currency into circulation (in billions per day).

Question 18

Find the center of mass of a lamina in the shape of a quarter-circle with radius 9 with density

Question 19

Find the centroid of the region shown,not by integration,but by locating the centroids of the rectangles and triangles and using additivity of moments.

Question 20

Biologists stocked a lake with 700 fish and estimated the carrying capacity (the maximal population for the fish of that species in that lake)to be 10800 .The number of fish tripled in the first year.Assuming that the size of the fish population satisfies the logistic equation,find an expression for the size of the population after t years.

Question 21

Suppose you make napkin rings by drilling holes with different diameters through two wooden balls (which also have different diameters).You discover that both napkin rings have the same height h as shown in the figure.Use cylindrical shells to compute the volume of a napkin ring created by drilling a hole with radius d through the center of a sphere of radius D and express the answer in terms of t.

A)
B)
C)
D)
E)

Question 22

Use the method of cylindrical shells to find the volume of solid obtained by rotating the region bounded by the given curves about the x-axis

Question 23

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line

A)
B)
C)
D)
E) None

Question 24

Find the arc length function for the curve with starting point .

A)
B)
C)
D)
E)

Question 25

Find the area of the surface obtained by rotating the curve about the x-axis.

A)
B)
C)
D)
E)

Question 26

Use the Midpoint Rule with n = 4 to estimate the volume obtained by rotating about the region under the y-axis the region under the curve. Select the correct answer.The choices are rounded to the nearest hundredth.

A)
B)
C)
D)
E)

Question 27

Find the area of the surface obtained by revolving the graph of y = on [0,1] about the x-axis.

Question 28

Find the area of the surface obtained by revolving the given curve about the y-axis. x = on [0,2]

Question 29

Find the center of mass of the system comprising masses m_k located at the points P_k in a coordinate plane.Assume that mass is measured in grams and distance is measured in centimeters.
m₁ = 3 , m₂ = 4, m₃ = 5
P₁ (-3,5), P₂ (3,4), P₃ (-4,1)

Question 30

Set up,but do not evaluate,an integral for the length of the curve

Question 31

A steady wind blows a kite due west.The kite's height above ground from horizontal position to ft is given by .
Find the distance traveled by the kite. Give your answer rounded to two decimal places.

Question 32

Find the area of the surface obtained by rotating the curve about the y-axis.

A)
B)
C)
D)
E) None of these

Question 33

Find the area of the surface obtained by rotating the circle about the line .

A)
B)
C)
D)
E)

Question 34

Use cylindrical shells to find the volume of the solid. A sphere of radius .

A)
B)
C)
D)
E)

Question 35

Find the volume of the solid obtained by rotating the region bounded by about the x-axis.

A)
B)
C)
D)
E)

Question 36

Use the arc length formula to find the length of the curve

Question 37

Use the method of cylindrical shells to find the volume of solid obtained by rotating the region bounded by the given curves about the x-axis.

A)
B)
C)
D)
E)

Question 38

Find the area of the surface obtained by rotating the curve about the x-axis.

Question 39

Find the volume of the solid obtained by rotating the region bounded by about the line

A)
B)
C)
D)
E)

Question 40

Set up,but do not evaluate,an integral for the area of the surface obtained by rotating the curve about the given axis. y-axis

A)
B)
C)
D)
E)

Question 41

The base of S is the parabolic region Cross-sections perpendicular to the y axis are squares. Find the volume of S.

Question 42

Find the values of c such that the area of the region bounded by the parabolas is 27.

Question 43

Find the volume of the solid obtained by rotating the region bounded by and about the line

Question 44

Find the area of the region bounded by the given curves.

A)
B)
C) 2
D)
E) 4

Question 45

Racing cars driven by Chris and Kelly are side by side at the start of a race.The table shows the velocities of each car (in miles per hour)during the first ten seconds of the race.Use the Midpoint Rule to estimate how much farther Kelly travels than Chris does during the first ten seconds.

Question 46

Find the volume of a cap of a sphere with radius r = 100 and height h = 39

Question 47

Find the number b such that the line divides the region bounded by the curves and into two regions with equal area.

A)
B)
C)
D)
E)

Question 48

Sketch the region enclosed by Find the area of the region.

Question 49

Find the area of the region bounded by the given curves.

A)
B)
C)
D)
E) None of these

Question 50

Find the volume of the frustum of a pyramid with square base of side b = 17, square top of side a = 5, and height h = 19.

Question 51

Sketch the region enclosed by the curves Find the area of the region correct to two decimal places.

A)
B)
C)
D)
E)

Question 52

The volume of a solid torus (the donut-shaped solid shown in the figure)with r = 5 and R = 15 is

Question 53

Find the area of the shaded region.

A)
B)
C)
D)

Question 54

Find the volume of a pyramid with height 4 and base an equilateral triangle with side a = 4.

A)
B)
C)
D)
E)

Question 55

The base of S is a circular region with boundary curve Cross-sections perpendicular to the x axis are isosceles right triangles with hypotenuse in the base. Find the volume of S.

Question 56

Find the volume of the solid obtained by rotating the region bounded by and about the y-axis.

Question 57

Find the volume common to two spheres,each with radius r = 12 if the center of each sphere lies on the surface of the other sphere.

Question 58

Find the area of the region bounded by the parabola ,the tangent line to this parabola at ,and the x axis.

A)
B)
C)
D)
E)

Question 59

Find the area of the shaded region.

A) 7
B)
C)
D)

Question 60

Use a computer algebra system to find the exact volume of the solid obtained by rotating the region bounded by the given curves about the specified line

Question 61

Sketch the region bounded by the graphs of the given equations and find the area of that region.
y = + 4,y = 2x + 1