Question 1

Find the length of the curve $x = \sin ^ { 3 } t , y = \cos ^ { 3 } t , 0 \leq t \leq 2 \pi$ .

Accepted Answer

A) 0 
B)  $\frac { 3 } { 2 }$ 
C) 2 
D)  $\pi$ 
E) 12
F)6
G)4

H)2 $\pi$ 
A) 0 
B)  $\frac { 3 } { 2 }$ 
C) 2 
D)  $\pi$ 
E) 12
F)6
G)4

H)2 $\pi$

Question 2

Find the area of the shaded region:

Accepted Answer

The answer of Find the area of the shaded region:...

Question 3

Find the center of mass of the linear system $m _ { 1 } = 2 , x _ { 1 } = - 3 , m _ { 2 } = 8$ $x _ { 2 } = - 1 , m _ { 3 } = 3$ $x _ { 3 } = 1 , m _ { 4 } = 5 , x _ { 4 } = 4$

Accepted Answer

A) 9 
B)  $\frac { 1 } { 2 }$ 
C)  $- \frac { 1 } { 2 }$ 
D) 0 
E)  $\frac { 2 } { 3 }$ 
F) $- \frac { 2 } { 3 }$ 
G) $- 9$ 
H)18 
A) 9 
B)  $\frac { 1 } { 2 }$ 
C)  $- \frac { 1 } { 2 }$ 
D) 0 
E)  $\frac { 2 } { 3 }$ 
F) $- \frac { 2 } { 3 }$ 
G) $- 9$ 
H)18

Question 4

Find the volume of the solid formed when the region bounded by the curves $y = x ^ { 3 } + 1 , x = 1 \text {, and } y = 0$ is rotated about the x-axis.

Accepted Answer

The answer of Find the volume of the solid formed...

Question 5

Find the area of the region bounded by the parabola $x = y ^ { 2 }$ and the line $x - 2 y = 3$ .

Accepted Answer

A)  $\frac { 29 } { 3 }$ 
B)  $\frac { 32 } { 3 }$ 
C)  $\frac { 35 } { 3 }$ 
D)  $\frac { 38 } { 3 }$ 
E)  $\frac { 41 } { 3 }$ 
F) $\frac { 44 } { 3 }$ 
G) $\frac { 47 } { 3 }$ 
H) $\frac { 50 } { 3 }$ 
A)  $\frac { 29 } { 3 }$ 
B)  $\frac { 32 } { 3 }$ 
C)  $\frac { 35 } { 3 }$ 
D)  $\frac { 38 } { 3 }$ 
E)  $\frac { 41 } { 3 }$ 
F) $\frac { 44 } { 3 }$ 
G) $\frac { 47 } { 3 }$ 
H) $\frac { 50 } { 3 }$

Question 6

Find the area of the region bounded by the curves $y = x ^ { 2 } - 4 x$ and $y = x - 4$

Accepted Answer

A)  $\frac { 8 } { 3 }$ 
B)  $\frac { 2 } { 3 }$ 
C)  $\frac { 1 } { 12 }$ 
D)  $\frac { 1 } { 2 }$ 
E)  $\frac { 1 } { 3 }$ 
F) $\frac { 1 } { 9 }$ 
G) $\frac { 9 } { 2 }$ 
H) $\frac { 5 } { 6 }$ 
A)  $\frac { 8 } { 3 }$ 
B)  $\frac { 2 } { 3 }$ 
C)  $\frac { 1 } { 12 }$ 
D)  $\frac { 1 } { 2 }$ 
E)  $\frac { 1 } { 3 }$ 
F) $\frac { 1 } { 9 }$ 
G) $\frac { 9 } { 2 }$ 
H) $\frac { 5 } { 6 }$

Question 7

Find the arc length of the curve $y = \ln ( \cos x ) , 0 \leq x \leq \frac { \pi } { 3 }$ .

Accepted Answer

A)  $\ln ( 2 + \sqrt { 2 } )$ 
B)  $\ln ( 1 + \sqrt { 3 } )$ 
C)  $\ln ( 2 - \sqrt { 2 } )$ 
D)  $\ln ( 2 - \sqrt { 3 } )$ 
E)  $\ln ( 1 + \sqrt { 2 } )$ 
F) $\ln ( 2 + \sqrt { 3 } )$ 
G) $\ln ( \sqrt { 3 } - 1 )$ 
H) $\ln ( \sqrt { 2 } - 1 )$ 
A)  $\ln ( 2 + \sqrt { 2 } )$ 
B)  $\ln ( 1 + \sqrt { 3 } )$ 
C)  $\ln ( 2 - \sqrt { 2 } )$ 
D)  $\ln ( 2 - \sqrt { 3 } )$ 
E)  $\ln ( 1 + \sqrt { 2 } )$ 
F) $\ln ( 2 + \sqrt { 3 } )$ 
G) $\ln ( \sqrt { 3 } - 1 )$ 
H) $\ln ( \sqrt { 2 } - 1 )$

Question 8

Give a definite integral representing the length of the parametric curve $x = t ^ { 3 } , y = t ^ { 4 } , 0 \leq t \leq 1$ .

Accepted Answer

A)  $\int _ { 0 } ^ { 1 } \left( t ^ { 3 } + t ^ { 4 } \right) d t$ 
B)  $\int _ { 0 } ^ { 1 } \sqrt { t ^ { 3 } + t ^ { 4 } } d t$ 
C)  $\int _ { 0 } ^ { 1 } \sqrt { 1 + 3 t ^ { 2 } } d t$ 
D)  $\int _ { 0 } ^ { 1 } \sqrt { t ^ { 2 } + 4 t ^ { 3 } } d t$ 
E)  $\int _ { 0 } ^ { 1 } \sqrt { 9 t ^ { 4 } + 16 t ^ { 6 } } d t$ 
F) $\int _ { 0 } ^ { 1 } \sqrt { 4 t ^ { 4 } + 9 t ^ { 6 } } d t$ 
G) $\int _ { 0 } ^ { 1 } \sqrt { t ^ { 5 } + t ^ { 7 } } d t$ 
H) $\int _ { 0 } ^ { 1 } \sqrt { 8 t ^ { 6 } + 6 t ^ { 8 } } d t$ 
A)  $\int _ { 0 } ^ { 1 } \left( t ^ { 3 } + t ^ { 4 } \right) d t$ 
B)  $\int _ { 0 } ^ { 1 } \sqrt { t ^ { 3 } + t ^ { 4 } } d t$ 
C)  $\int _ { 0 } ^ { 1 } \sqrt { 1 + 3 t ^ { 2 } } d t$ 
D)  $\int _ { 0 } ^ { 1 } \sqrt { t ^ { 2 } + 4 t ^ { 3 } } d t$ 
E)  $\int _ { 0 } ^ { 1 } \sqrt { 9 t ^ { 4 } + 16 t ^ { 6 } } d t$ 
F) $\int _ { 0 } ^ { 1 } \sqrt { 4 t ^ { 4 } + 9 t ^ { 6 } } d t$ 
G) $\int _ { 0 } ^ { 1 } \sqrt { t ^ { 5 } + t ^ { 7 } } d t$ 
H) $\int _ { 0 } ^ { 1 } \sqrt { 8 t ^ { 6 } + 6 t ^ { 8 } } d t$

Question 9

The temperature (in °C) of a metal rod 5 m long is 4x at a distance x meters from one end of the rod. What is the average temperature of the rod?

Accepted Answer

The answer of The temperature (in °C) of a metal...

Question 10

Find the arc length of the curve $x = t \cos t - \sin t , y = t \sin t + \cos t , 0 \leq t \leq \pi$ .

Accepted Answer

The answer of Find the arc length of the curve...

Question 11

Give a definite integral representing the length of the curve $y = \frac { 1 } { x } , 1 \leq x \leq 2$ .

Accepted Answer

A)  $\int _ { 1 } ^ { 2 } \sqrt { 1 + \frac { 1 } { x ^ { 2 } } } d x$ 
B)  $\int _ { 1 } ^ { 2 } \sqrt { 1 + \frac { 1 } { x ^ { 4 } } } d x$ 
C)  $\int _ { 1 } ^ { 2 } \sqrt { 1 + ( \ln x ) ^ { 2 } } d x$ 
D)  $\int _ { 1 } ^ { 2 } \sqrt { 1 + \frac { 1 } { x } } d x$ 
E)  $\int _ { 1 } ^ { 2 } \frac { 1 } { x ^ { 2 } } d x$ 
F) $\int _ { 1 } ^ { 4 } \sqrt { 1 + \frac { 1 } { x ^ { 4 } } } d x$ 
G) $\int _ { 1 } ^ { 2 } x \sqrt { 1 + \frac { 1 } { x ^ { 2 } } } d x$ 
H) $\int _ { 1 } ^ { 4 } \frac { 1 } { x ^ { 4 } } d x$ 
A)  $\int _ { 1 } ^ { 2 } \sqrt { 1 + \frac { 1 } { x ^ { 2 } } } d x$ 
B)  $\int _ { 1 } ^ { 2 } \sqrt { 1 + \frac { 1 } { x ^ { 4 } } } d x$ 
C)  $\int _ { 1 } ^ { 2 } \sqrt { 1 + ( \ln x ) ^ { 2 } } d x$ 
D)  $\int _ { 1 } ^ { 2 } \sqrt { 1 + \frac { 1 } { x } } d x$ 
E)  $\int _ { 1 } ^ { 2 } \frac { 1 } { x ^ { 2 } } d x$ 
F) $\int _ { 1 } ^ { 4 } \sqrt { 1 + \frac { 1 } { x ^ { 4 } } } d x$ 
G) $\int _ { 1 } ^ { 2 } x \sqrt { 1 + \frac { 1 } { x ^ { 2 } } } d x$ 
H) $\int _ { 1 } ^ { 4 } \frac { 1 } { x ^ { 4 } } d x$

Question 12

Find the volume of the solid generated by rotating the region bounded by $y = \frac { 1 } { x } , x = 1$ and the x-axis about the x-axis.

Accepted Answer

The answer of Find the volume of the solid generated...

Question 13

A tank 5 feet long has cross-sections n the shape of a parabola $y = x ^ { 2 } , \text { for } - 2 \leq x \leq 2$ (where x and y are in feet). Suppose that the tank is filled to a depth of 3 feet with liquid weighing 15 lb/ft3. How much work is required to empty the tank by pumping the liquid over the edge of the tank?

Accepted Answer

Question 14

A gate in an irrigation canal is in the form of a trapezoid 3 feet wide at the bottom, 5 feet wide at the top, with height equal to 2 feet. It is placed vertically in the canal with the water extending to its top. For simplicity, take the density of water to be 60 lb/ft3. Find the hydrostatic force in pounds on the gate.

Accepted Answer

A) 360 
B) 380 
C) 440 
D) 420 
E) 400
F)460
G)500
H)480 
A) 360 
B) 380 
C) 440 
D) 420 
E) 400
F)460
G)500
H)480

Question 15

Find the work (in ft-lb) done in raising 500 lb of ore from a mine that is 1000 ft deep. Assume that the cable used to raise the ore weighs 2 lb/ft.

Accepted Answer

A)  $5 \times 10 ^ { 5 }$ 
B)  $2.5 \times 10 ^ { 6 }$ 
C)  $2 \times 10 ^ { 6 }$ 
D)  $6 \times 10 ^ { 5 }$ 
E) 106
F) $1.5 \times 10 ^ { 6 }$ 
G) $7 \times 10 ^ { 5 }$ 
H) $4 \times 10 ^ { 5 }$ 
A)  $5 \times 10 ^ { 5 }$ 
B)  $2.5 \times 10 ^ { 6 }$ 
C)  $2 \times 10 ^ { 6 }$ 
D)  $6 \times 10 ^ { 5 }$ 
E) 106
F) $1.5 \times 10 ^ { 6 }$ 
G) $7 \times 10 ^ { 5 }$ 
H) $4 \times 10 ^ { 5 }$

Question 16

The demand function for a certain commodity is $p ( x ) = \frac { 1800 } { ( x + 5 ) ^ { 2 } }$ , and consumer surplus is 90. What should be the selling price?

Accepted Answer

Solve for

Question 17

A culture of bacteria is doubling every hour. What is the average population over the first two hours if we assume that the culture initially contained two million organisms?

Accepted Answer

The answer of A culture of bacteria is doubling every...

Question 18

A right circular cylinder tank of height 1 foot and radius 1 foot is full of water. Taking the density of water to be a nice round 60 pounds per cubic foot, how much work in foot-pounds is required to pump all of the water up and over the top of the tank?

Accepted Answer

A) 8 $\pi$ 
B) 24 $\pi$ 
C) 16 $\pi$ 
D) 6 $\pi$ 
E) 5 $\pi$ 
F)4 $\pi$ 
G)30 $\pi$ 
H)18 $\pi$ 
A) 8 $\pi$ 
B) 24 $\pi$ 
C) 16 $\pi$ 
D) 6 $\pi$ 
E) 5 $\pi$ 
F)4 $\pi$ 
G)30 $\pi$ 
H)18 $\pi$

Question 19

Let R be the region bounded by: $y = x ^ { 3 }$ , the tangent to $y = x ^ { 3 }$ at $( 1,1 )$ , and the x-axis.Find the area of R integrating
(a) with respect to x.(b) with respect to y.

Accepted Answer

The answer of Let R be the region bounded by:...

Question 20

The marginal revenue from selling x items is $90 - 0.02 x$ . The revenue from the sale of the first 100 items is $8800. What is the revenue from the sale of the first 200 items?

Accepted Answer

The answer of The marginal revenue from selling x items...

Find the length of the curve $x = \sin ^ { 3 } t , y = \cos ^ { 3 } t , 0 \leq t \leq 2 \pi$ .

Find the area of the shaded region:

Find the center of mass of the linear system $m _ { 1 } = 2 , x _ { 1 } = - 3 , m _ { 2 } = 8$ $x _ { 2 } = - 1 , m _ { 3 } = 3$ $x _ { 3 } = 1 , m _ { 4 } = 5 , x _ { 4 } = 4$

Find the volume of the solid formed when the region bounded by the curves $y = x ^ { 3 } + 1 , x = 1 \text {, and } y = 0$ is rotated about the x-axis.

Find the area of the region bounded by the parabola $x = y ^ { 2 }$ and the line $x - 2 y = 3$ .

Find the area of the region bounded by the curves $y = x ^ { 2 } - 4 x$ and $y = x - 4$

Find the arc length of the curve $y = \ln ( \cos x ) , 0 \leq x \leq \frac { \pi } { 3 }$ .

Give a definite integral representing the length of the parametric curve $x = t ^ { 3 } , y = t ^ { 4 } , 0 \leq t \leq 1$ .

The temperature (in °C) of a metal rod 5 m long is 4x at a distance x meters from one end of the rod. What is the average temperature of the rod?

Find the arc length of the curve $x = t \cos t - \sin t , y = t \sin t + \cos t , 0 \leq t \leq \pi$ .

Give a definite integral representing the length of the curve $y = \frac { 1 } { x } , 1 \leq x \leq 2$ .

Find the volume of the solid generated by rotating the region bounded by $y = \frac { 1 } { x } , x = 1$ and the x-axis about the x-axis.

Find the work (in ft-lb) done in raising 500 lb of ore from a mine that is 1000 ft deep. Assume that the cable used to raise the ore weighs 2 lb/ft.

The demand function for a certain commodity is $p ( x ) = \frac { 1800 } { ( x + 5 ) ^ { 2 } }$ , and consumer surplus is 90. What should be the selling price?

A culture of bacteria is doubling every hour. What is the average population over the first two hours if we assume that the culture initially contained two million organisms?

A right circular cylinder tank of height 1 foot and radius 1 foot is full of water. Taking the density of water to be a nice round 60 pounds per cubic foot, how much work in foot-pounds is required to pump all of the water up and over the top of the tank?

Let R be the region bounded by: $y = x ^ { 3 }$ , the tangent to $y = x ^ { 3 }$ at $( 1,1 )$ , and the x-axis.Find the area of R integrating (a) with respect to x.(b) with respect to y.

The marginal revenue from selling x items is $90 - 0.02 x$ . The revenue from the sale of the first 100 items is $8800. What is the revenue from the sale of the first 200 items?

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Exam 6: Applications of Integration

Find the length of the curve x=sin⁡3t,y=cos⁡3t,0≤t≤2πx = \sin ^ { 3 } t , y = \cos ^ { 3 } t , 0 \leq t \leq 2 \pix=sin3t,y=cos3t,0≤t≤2π .

Find the area of the shaded region:

Find the center of mass of the linear system m1=2,x1=−3,m2=8m _ { 1 } = 2 , x _ { 1 } = - 3 , m _ { 2 } = 8m1​=2,x1​=−3,m2​=8 x2=−1,m3=3x _ { 2 } = - 1 , m _ { 3 } = 3x2​=−1,m3​=3 x3=1,m4=5,x4=4x _ { 3 } = 1 , m _ { 4 } = 5 , x _ { 4 } = 4x3​=1,m4​=5,x4​=4

Find the volume of the solid formed when the region bounded by the curves y=x3+1,x=1, and y=0y = x ^ { 3 } + 1 , x = 1 \text {, and } y = 0y=x3+1,x=1, and y=0 is rotated about the x-axis.

Find the area of the region bounded by the parabola x=y2x = y ^ { 2 }x=y2 and the line x−2y=3x - 2 y = 3x−2y=3 .

Find the area of the region bounded by the curves y=x2−4xy = x ^ { 2 } - 4 xy=x2−4x and y=x−4y = x - 4y=x−4

Find the arc length of the curve y=ln⁡(cos⁡x),0≤x≤π3y = \ln ( \cos x ) , 0 \leq x \leq \frac { \pi } { 3 }y=ln(cosx),0≤x≤3π​ .

Give a definite integral representing the length of the parametric curve x=t3,y=t4,0≤t≤1x = t ^ { 3 } , y = t ^ { 4 } , 0 \leq t \leq 1x=t3,y=t4,0≤t≤1 .

The temperature (in °C) of a metal rod 5 m long is 4x at a distance x meters from one end of the rod. What is the average temperature of the rod?

Find the arc length of the curve x=tcos⁡t−sin⁡t,y=tsin⁡t+cos⁡t,0≤t≤πx = t \cos t - \sin t , y = t \sin t + \cos t , 0 \leq t \leq \pix=tcost−sint,y=tsint+cost,0≤t≤π .

Give a definite integral representing the length of the curve y=1x,1≤x≤2y = \frac { 1 } { x } , 1 \leq x \leq 2y=x1​,1≤x≤2 .

Find the volume of the solid generated by rotating the region bounded by y=1x,x=1y = \frac { 1 } { x } , x = 1y=x1​,x=1 and the x-axis about the x-axis.

Find the work (in ft-lb) done in raising 500 lb of ore from a mine that is 1000 ft deep. Assume that the cable used to raise the ore weighs 2 lb/ft.

The demand function for a certain commodity is p(x)=1800(x+5)2p ( x ) = \frac { 1800 } { ( x + 5 ) ^ { 2 } }p(x)=(x+5)21800​ , and consumer surplus is 90. What should be the selling price?

A culture of bacteria is doubling every hour. What is the average population over the first two hours if we assume that the culture initially contained two million organisms?

A right circular cylinder tank of height 1 foot and radius 1 foot is full of water. Taking the density of water to be a nice round 60 pounds per cubic foot, how much work in foot-pounds is required to pump all of the water up and over the top of the tank?

Let R be the region bounded by: y=x3y = x ^ { 3 }y=x3 , the tangent to y=x3y = x ^ { 3 }y=x3 at (1,1)( 1,1 )(1,1) , and the x-axis.Find the area of R integrating (a) with respect to x.(b) with respect to y.

The marginal revenue from selling x items is 90−0.02x90 - 0.02 x90−0.02x . The revenue from the sale of the first 100 items is $8800. What is the revenue from the sale of the first 200 items?

Functions and Models

Limits and Derivatives

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Applications of Differentiation

Integrals

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Vector Calculus

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Find the length of the curve $x = \sin ^ { 3 } t , y = \cos ^ { 3 } t , 0 \leq t \leq 2 \pi$ .

Find the center of mass of the linear system $m _ { 1 } = 2 , x _ { 1 } = - 3 , m _ { 2 } = 8$ $x _ { 2 } = - 1 , m _ { 3 } = 3$ $x _ { 3 } = 1 , m _ { 4 } = 5 , x _ { 4 } = 4$

Find the volume of the solid formed when the region bounded by the curves $y = x ^ { 3 } + 1 , x = 1 \text {, and } y = 0$ is rotated about the x-axis.

Find the area of the region bounded by the parabola $x = y ^ { 2 }$ and the line $x - 2 y = 3$ .

Find the area of the region bounded by the curves $y = x ^ { 2 } - 4 x$ and $y = x - 4$

Find the arc length of the curve $y = \ln ( \cos x ) , 0 \leq x \leq \frac { \pi } { 3 }$ .

Give a definite integral representing the length of the parametric curve $x = t ^ { 3 } , y = t ^ { 4 } , 0 \leq t \leq 1$ .

Find the arc length of the curve $x = t \cos t - \sin t , y = t \sin t + \cos t , 0 \leq t \leq \pi$ .

Give a definite integral representing the length of the curve $y = \frac { 1 } { x } , 1 \leq x \leq 2$ .

Find the volume of the solid generated by rotating the region bounded by $y = \frac { 1 } { x } , x = 1$ and the x-axis about the x-axis.

The demand function for a certain commodity is $p ( x ) = \frac { 1800 } { ( x + 5 ) ^ { 2 } }$ , and consumer surplus is 90. What should be the selling price?

Let R be the region bounded by: $y = x ^ { 3 }$ , the tangent to $y = x ^ { 3 }$ at $( 1,1 )$ , and the x-axis.Find the area of R integrating (a) with respect to x.(b) with respect to y.

The marginal revenue from selling x items is $90 - 0.02 x$ . The revenue from the sale of the first 100 items is $8800. What is the revenue from the sale of the first 200 items?