Fun with Areas
The Revolution will not be Televised
Washers or Shells
I've Been Working on the Railroad
Potpourri
100

The integral used to find the area bounded by y = 2^x , y = 8, and the y-axis. (Must include a sketch of the region)

What is 

int_0^3 (8-2^x)dx

100

The integral used to compute the volume of the solid obtained by rotating the region enclosed by y = x^3 and y = 4x about the x-axis.

What is 

pi int_0^2 (16x^2-x^6) dx = (512pi)/21

100

The integral used to find the volume of the solid obtained by rotating the region bounded by the graphs of y = sqrt(x), y = 2 − x and y = 0 around the y-axis. (You must sketch the region.)

pi int_0^1 [(2-y)^2-(y^2)^2]dy

100

Suppose a force of 10 N is required to stretch a spring 0.1 m from its equilibrium position and hold it in that position. How much work is needed to stretch the spring 0.25 m from its equilibrium position?

What is 3.125 J

100

The arc length of

y = ln(secx)

 for 

[0,pi/4]

What is

ln(sqrt(2) + 1)

200

The integral used to find the area bounded by y = 3^x , x = 2, the x-axis, and the y-axis. (Must include a sketch of the region.)

What is Int(3^x,x,0,2)

int_0^2 3^xdx

200

The integral used to compute the volume of the solid obtained by rotating the region enclosed by y = x^3 and y = 4x about y = -1.

What is

pi int_0^2 [(4x + 1)^2 - (x^3 + 1)^2]dx

200

The integral used to find the volume obtained by revolving the region bounded by y = x^2 − 4 and y = 4 − x^2 around the line x = 2? (You must sketch the region.)

What is 

2pi int_-2^2[(2 - x)((4 - x^2) - (x^2 - 4))]dx

200
Archaeologists have determined that the Great Statue of Aruba was really a giant magnet placed on top of an iron table. At a height of x feet above the table, the magnetic force exerted on the statue was given by F(x) = 1600/(2x + 1)^2 lb. When the mighty Hercules lifted the statue 3 feet before hurling it at Ares, how much work did he do? (You may ignore gravity; the statue was hollow and had very little mass.)
What is 4800/7 ft-lbs
200

Set up an integral to find the surface area formed by rotating f(x) = sin(x), on the interval [0,pi] about the x-axis.

What is

2pi int_0^pi sin(x)sqrt(1 + cos^2(x))dx

300

The integral used to find the area in the first quadrant between x^2 + y^2 = 1 and x^(1/2) + y^(1/2) = 1. (Must include a sketch of the region.)

What is 

int_0^1 (sqrt(1-x^2) - (1 - sqrt(x))^2)dx

300

The integral used to compute the volume of the solid obtained by rotating the region enclosed by y = x^3 and y = 4x about y = 9

What is 

pi int_0^2 [(9 - x^3)^2 - (9 - 4x)^2]dx

300

The integral used to find the volume of the solid obtained by rotating the area between the graphs of y = x^2 and x = 2y around the y-axis. (You must sketch the region.)

What is Shell: 

2pi int_0^(1/2) x(x/2 - x^2)dx

Washer: 

pi int_0^(1/4) [(sqrt(y))^2-(2y)^2]dy

300
A bucket that weighs 70 lb when filled with water is lifted at a constant rate, by a mechanical winch, from the bottom of a well that is 60 feet deep. The chain that is being used to lift the bucket weighs 0.55 pounds per foot. Now find the amount of work required to lift the bucket from the bottom of the well all the way to the top.
What is 5,190 ft-lbs
300

Find the center of mass for the region enclosed by

y = 4 - x^2

 and the x-axis.

What is

(0,8/5)

400

The integral used to find the area between the curves y = cos(x) and y = 1/2 x - 1, bounded on the left by the y-axis. Use technology to find the point of intersection rounded to 3 decimal places. (Must include a sketch of the graph.)

What is 

int_0^1.714 (cosx - (1/2 x - 1))dx

400

The integral used to compute the volume of the solid obtained by rotating the region enclosed by y = x^3 and y = 4x about the y-axis.

What is 

pi int_0^8 [y^(2/3) - (y/4)^2]dy

400

Integral used to find the volume of the solid obtained by rotating the region between the graphs of y = x sqrt(2 - x) and y = 0 around the x-axis. (You must sketch the region.)

What is 

pi int_0^2 x^2(2-x)dx

400
It takes 100 J of work to stretch a spring 0.5 m from its equilibrium position. How much work is needed to stretch it an additional 0.75 m?
What is 525 J
500

The integral used to find The area bounded by the curves y = x^2 - 4 and y ={ 1/2 x + 1 if x <= 0, -1/2 x + 1 if x > 0 (Must include sketch of graph)

What is Int((1/2 x + 1) - (x^2 - 4),x,-2,0) + Int((-1/2 x + 1) - (x^2 - 4),x,0,2)

int_-2^0 [(1/2 x+1)-(x^2-4)]dx+int_0^2 [(-1/2 x+1)-(x^2-4)]dx

500

The integral used to compute the volume of the solid obtained by rotating the region enclosed by y = x^3 and y = 4x about x = 2.

What is 

pi int _0^8 [(2 - y/4)^2 - (2 - y^(1/3))^2]dy

500

The Integral used to find the volume formed by rotating the region enclosed by y = 2x - x^2 and the x-axis about the line x = -1. (You must sketch the region.)

What is 

2pi int_0^2(x+1)(2x - x^2)dx

500
A cylindrical tank with a length of 10 m and a radius of 5 m is on its side and half-full of gasoline. How much work is required to empty the tank through an outlet pipe at the top of the tank? (The density of gasoline is 737 kg/m^3)
What is 20.2 million J
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