Deck 6: Applications of the Definite Integral in Geometry, Science and Engineering

Question 1

Evaluate sinh(7).

A) 0.6570
B) 1.5221
C) 548.3161
D) 0.1433
E) 49

Answer 1

548.3161

Question 2

Find if y = sinh(5x²).

A) 5x² cosh(5x²)
B) 10x cosh(10x)
C) 10x sinh(5x²)
D) 10x cosh(5x²)
E) 5x² sinh(5x²)

Answer 2

10x cosh(5x²)

Question 3

$\int$ cosh⁷x sinh x dx =

A) 8 cosh⁸x + C
B) cosh⁸x + C
C)
D) 7 cosh⁸x + C
E)

Answer 3

Question 4

$\int$ sinh(7x + 8) dx =

A)
B)
C)
D)
E)

Question 5

Evaluate cosh ^{- 1}[cosh(-6)].

Question 6

Answer true or false. = 0

Question 7

Find f '(x) if f(x) = tanh ^{- 1}(6x + 9).

Question 8

Answer true or false. = sinh^-1 e^x + C

Question 9

Evaluate cosh(6 sinh ^{- 1}8).

Question 10

Find if .

A)
B)
C)
D)
E)

Question 11

Simplify sinh (cosh ^{- 1}6).

Question 12

Find f '(x) if f(x) = sinh ^{- 1}(6x - 5).

Question 13

Answer true or false. =

Question 14

Find if .

A)
B)
C)
D)
E)

Question 15

Evaluate cosh^-1(7).

A) 0.7539
B) 2.7687
C) 2.4779
D) 2.6441
E) 2.6339

Question 16

Find if y = sinh(8x + 1).

A) (8x + 1)cosh(8x + 1)
B) 8 cosh(8x + 1)
C) cosh(8)
D) (8x + 1)sinh(8x + 1)
E) 8 sinh(8x + 1)

Question 17

Answer true or false. If when |x| > 0, .

Question 18

Answer true or false. = 1

Question 19

Evaluate cosh(sinh ^{- 1}24).

Question 20

Evaluate cosh(cosh ^{- 1}15).

Question 21

Find f '(x) if .

Question 22

Answer true or false. The force a liquid of density $\rho$ exerts on an equilateral triangle with edges h in length submerged point down is given by .

Question 23

Evaluate .

Question 24

Find f'(x) if f(x) = (sinh ^{- 1}x)^x .

Question 25

Find f '(x) if .

Question 26

A flat rectangular plate is submerged horizontally in water to a depth of 3.6 ft. If the top surface of the plate has an area of 30 ft², and the liquid in which it is submerged is water, find the force on the top of the plate. Neglect the effect of the atmosphere above the liquid. (The density of water is 62.2 lb/ft³.)

A) 108 lb
B) 22,392 lb
C) 1,866 lb
D) 6,717.6 lb
E) 671.76 lb

Question 27

Evaluate .

Question 28

Answer true or false. A semicircular wall 8 ft across the top forms one vertical end of a tank. The total force exerted on this wall by water, if the tank is full of water, is 12,546.26 lb. Neglect the effect of the atmosphere above the liquid. (The weight density of water is 62.4 lb/ft³.)

Question 29

Find f '(x) if f(x) = sinh ^{- 1}(sin 9x).

Question 30

A right triangle is submerged vertically with one side at the surface in a liquid of density $\rho$ . The triangle has a leg that is 9 m long located at the surface and a leg 9 m long straight down. Find the force exerted on the triangular surface, in terms of the density. Neglect the effect of the atmosphere above the liquid.

A) 243 $\rho$ N
B) 364.5 $\rho$ N
C) 121.5 $\rho$ N
D) 91.12 $\rho$ N
E) 20.25 $\rho$ N

Question 31

A right triangle is submerged vertically with one side parallel to the surface in a liquid of density $\rho$ . The triangle has a leg that is 12 m long located at the surface and a leg 12 m long straight down. Find the force exerted on the triangular surface, in terms of the density. Neglect the effect of the atmosphere above the liquid.

A) 288 $\rho$ N
B) 864 $\rho$ N
C) 576 $\rho$ N
D) 48 $\rho$ N
E) 36 $\rho$ N

Question 32

Answer true or false. A 12 in diameter glass circular window on the side of a submarine has the same force acting on the top half as on the bottom half.

Question 33

Find the force (in N) on the top of a submerged object if its surface is 4.2 m² and the pressure acting on it is 6.6 * 10⁶ Pa. Neglect the effect of the atmosphere above the liquid.

A) 6,600,000 N
B) 27,720,000 N
C) 4,200,000 N
D) 166.32 N
E) 66,000,000 N

Question 34

Evaluate .

Question 35

Find f '(x) if f(x) = (tanh ^{- 1}2x)².

Question 36

Answer true or false. If a completely full conical tank is inverted, the total force on the side wall will be the same.

Question 37

Find the force on a 44-ft wide by 5-ft deep wall of a swimming pool filled with water. Neglect the effect of the atmosphere above the liquid. (The weight density of water is 62.4 lb/ft³.)

A) 13,728 lb
B) 68,640 lb
C) 302,016 lb
D) 24,200 lb
E) 34,320 lb

Question 38

Evaluate .

Question 39

Find f '(x) if .

Question 40

Find the force on a 190 ft² horizontal surface 22 ft deep in water. Neglect the effect of the atmosphere above the liquid. (The density of water is 62.4 lb/ft³.)

A) 4,180 lb
B) 260,832 lb
C) 130,416 lb
D) 8,346,624 lb
E) 9,810 lb

Question 41

A right triangle is submerged vertically with one side at the surface in a liquid of density $\rho$ . The triangle has a leg that is 5 m long located at the surface and a leg 5 m long straight down. Find the force exerted on the triangular surface, in terms of the density. Neglect the effect of the atmosphere above the liquid.

Question 42

Answer true or false. If a square, flat surface is suspended vertically in water and its center is 64 m deep, the force on the bottom side of the object will triple if the object is relocated to a depth of 192 m. Neglect the effect of the atmosphere above the liquid.

Question 43

Answer true or false. The force on a semicircular, vertical wall with top d is given by . Neglect the effect of the atmosphere above the liquid and assume the liquid is at a depth equal to the height of the wall. (The letter stands for the weight of the liquid.)

Question 44

Find the force (in N) on the top of a submerged object if its surface is 6.6 m² and the pressure acting on it is 4.6 * 10⁴ Pa. Neglect the effect of the atmosphere above the liquid.

Question 45

Liquid cement (weight density 250lbs/ft³) is poured into a form whose ends are 3 foot squares. Find the force on one end if the cement is 2 feet deep.

Question 46

A flat plate, shaped in the form of a semicircle 10 meters in diameter is submerged in water (weight density 9810 N/m³) as shown. Find the force against the surface of the plate.

Question 47

A flat triangular plate whose dimensions are 17, 17, and 30 feet is submerged in water (weight density 62.4 lbs/ft³) point up so that its longer side is below the surface and parallel to it and its vertex is 4 feet below the water. Find the force against the surface of the plate.

Question 48

A semicircular wall 10 ft across the top forms one vertical end of a tank. What is the total force exerted on this wall by water if the tank is full of water? Neglect the effect of the atmosphere above the liquid. (The weight density of water is 62.4 lb/ft³.)

Question 49

Answer true or false. If a submerged rectangle is rotated 125° about an axis through its center and perpendicular to its surface, the force exerted on one side of it will be the same, provided the entire rectangle remains submerged.

Question 50

A flat rectangular plate is submerged horizontally in water to a depth of 3.2 ft. If the top surface of the plate has an area of 50 ft², and the liquid in which it is submerged is water, find the force on the top of the plate. Neglect the effect of the atmosphere above the liquid. (The density of water is 62.6 lb/ft³.)

Question 51

Answer true or false. The force exerted by water on a surface of a square, vertical plate (suspended vertically) with edges of 4 ft if it is suspended with its top 4 ft below the surface is 5,990.4 lb. (The density of water is 62.4 lb/ft³.)

Question 52

A right triangle is submerged vertically with one side parallel to the surface in a liquid of density $\rho$ . The triangle has a leg that is 16 m long located at the surface and a leg 16 m long straight down. Find the force exerted on the triangular surface, in terms of the density. Neglect the effect of the atmosphere above the liquid.

Question 53

Answer true or false. A 106 ft² flat sheet of material is submerged vertically in water. The force acting on each side must be the same.

Question 54

A flat rectangular plate, 5 feet long and 6 feet wide is submerged in water (weight density 62.4 lbs/ft³) with the 5 foot edge parallel to and 3 feet below the surface. Find the force against the surface of the plate.

Question 55

Find the force on a 30-ft wide by 4-ft deep wall of a swimming pool filled with water. Neglect the effect of the atmosphere above the liquid. (The density of water is 62.4 lb/ft³.)

Question 56

A horizontal cylindrical tank of diameter 12 feet is half full of a chemical (weight density 55 lbs/ft³). Calculate the force against one end.

Question 57

What is the force exerted by water on a surface of a square, vertical plate with edges of 3 ft if it is suspended with its top 3 ft below the surface? (The density of water is 62.4 lb/ft³.)

Question 58

Answer true or false. If a submerged horizontal object is elevated to twice its original depth, the force exerted on the top of the object will be twice the force originally exerted on the object. Assume there is a vacuum above the liquid surface.

Question 59

Find the force on a 60 ft² horizontal surface 48 ft deep in water. Neglect the effect of the atmosphere above the liquid. (The density of water is 62.4 lb/ft³.)

Question 60

A flat triangular plate whose dimensions are 13, 13, and 10 feet is submerged in water (weight density 62.4 lbs/ft³) so that its longer side is at the surface and parallel to it. Find the force against the surface of the plate.

Question 61

Find the centroid of a triangle with vertices (0, 0), (2, 0), and (0, 6).

Question 62

A trapezoidal gate in a dam that is 5 feet wide at the top, 3 feet wide at the bottom, and 4 feet tall is submerged in water (weight density 62.4 lbs/ft³) as shown in the figure. Find the force on the gate when the surface of the water is 2 feet above the top of the gate.

Question 63

Find the Mass of a lamina bounded by the x-axis, the line x = 4, and the curve y = x³. The density of the lamina is = 4.

Question 64

Find the Mass of a lamina bounded by the graph of y = x² and y = 4. The density of the lamina is = 5.

Question 65

A swimming pool is 24 feet long and 16 feet wide. The bottom is flat but inclined as shown in the figure. The water is 12 feet deep on one end and 2 feet deep on the other. Find the force on the bottom of the pool when it is filled with water (weight density 62.4 lbs/ft³).

Question 66

Find the centroid of a triangle with vertices (0, 0), (8, 0), and (0, 3).

A) (1.5, 4)
B) (1, 2.666667)
C) (2.666667, 1)
D) (4, 1.5)
E) (2.666667, 0)

Question 67

Find the centroid of the region enclosed by y = -x² + 8 and from x = 0 to x = 1.

Question 68

A swimming pool is 35 feet long and 20 feet wide. The bottom is flat but inclined as shown in the figure. The water is 7 feet deep on one end and 3 feet deep on the other. Find the force on one of the sides when the pool is filled with water (weight density 62.4 lbs/ft³).

Question 69

Find the Mass of a lamina bounded by the y-axis and the graph of x = 81 - y⁴. The density of the lamina is = 3.

A) 388.8
B) 583.2
C) 1166.4
D) 1184.7
E) 1148.1

Question 70

Masses m₁ = 6, m₂ = 3, m₃ = 11 are positioned on a weightless on a weightless beam as shown in the figure. Where should be fulcrum be placed so that the beam is in equilibrium?

Question 71

A flat plate in the form of a T is submerged in water (weight density 9810 N/m³) as shown in the figure. The cross piece is 10 meters long (horizontally) and 3 meters wide (deep). The upright is 14 meters long (vertically) and 3 meters wide. Find the force on the surface when the surface of the water is 3 meters above the cross piece.

Question 72

The face of the dam shown in the figure is an inclined rectangle. If the dam is 60 feet wide, the face is 30 feet in length and $\theta$ = 30°, find the fluid force on the face when the water (weight density 62.4 lbs/ft³) is level with the top of the dam.

Question 73

Find the centroid of the region enclosed by the graphs of , x = y , and y = 4.

Question 74

Find the centroid of the region enclosed by y = x³ - 2 and y = 3x + 7 from x = -1 to x = 0.

A) (-0.477419, 1.668203)
B) (0.425806, 1.840246)
C) (1.840246, -0.477419)
D) (-0.477419, 1.840246)
E) (0.425806, 1.668203)

Question 75

Find the centroid of the region bounded by the graphs of xy = 6 and x + y = 5.

A) (2.479814, -2.479814)
B) (2.479814, 34.833333)
C) (2.479814, 2.479814)
D) (-2.479814, 2.479814)
E) (1.779814, 2.679814)

Question 76

Find the centroid of the region bound by y = x + 3 and y = x² - 3.

A) (0, 1.68)
B) (1.098, 1.68)
C) (0, 4)
D) (1.68, 1.098)
E) (0, 2.18)

Question 77

Find the fluid pressure on the top of a rectangular plate submerged in water at a depth of 2 m. (The weight density of water is $\rho$ = 9,810 N/m³.)

Question 78

A trapezoidal gate in a dam that is 2 feet wide at the top, 4 feet wide at the bottom, and 6 feet tall is submerged in water (weight density 62.4 lbs/ft³) as shown in the figure. Find the force on the gate when the surface of the water is 2 feet above the top of the gate.

Question 79

Find the centroid of the region bound by y = x + 2 and y = x² - 4.

Question 80

Masses m₁ = 7, m₂ = 9, m₃ = 11 are positioned on a weightless on a weightless beam as shown in the figure. Where should be fulcrum be placed so that the beam is in equilibrium?

A)
B)
C)
D) 11
E)