CALC
Warm Up: Review of Integration by Substitution
Area of a Surface of Revolution
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Find the indefinite integral by using u-substitution.
1.
x
3.
 4x
3x 2  2dx
2
2.
x
3
1  x 4 dx
(2 x 3  1) 6 dx
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4. Find the area of the surface formed by revolving the graph of y  x 3 on the interval [0, 3]
3
about the x-axis.
CALC
7 – 5 Work Done by a Constant Force
Important concept for scientists and engineers to determine energy needed to perform various
jobs (cranes lifting, trucks pulling, rockets launching, springs compressing/stretching, etc)
Work is done by a force when it moves an object. A force can be thought of as a push or pull; a
force changes the state of rest or state of motion of a body. For gravitational forces on Earth, it
is common to use units of measure corresponding to the weight of an object.
If an object is moved a distance D in the direction of an applied constant force, F, then the
work W done by the force is defined as W = FD
1. Determine the work done by lifting a 50 lb object 4 feet.
Work is typically expressed in foot-pounds (ft-lbs), inch-pounds, or foot-tons. In centimetergram-seconds (C-G-S) the basic unit is a dyne, so we can answer in dyne-cm (ergs) or newtonmeters (joules) as well.
Work Done by a Variable Force – Hooke’s Law
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If a variable force is applied to an object, calculus is needed to determine the work done,
because the amount of force changes as the object changes position. For instance, the force
required to compress a spring increases as the spring is compressed.
Definition of Work Done by a Variable Force:
If an object is moved along a straight line by a continuously varying force F(x), then the work W
done by the force as the object is moved from x = a to x = b is
b
W   F ( x)dx
a
Hooke’s Law: developed by Robert Hooke (1635-1703)
The force F required to compress or stretch a spring (within its elastic limits) is proportional to
the distance d that the spring is compressed or stretched from its original length. That is,
F = kd
where k = the constant of proportionality (the spring constant)
since the distance is variable – we can use x!
1. A force of 250 newtons stretches a spring 20 centimeters. How much work is done in
stretching the spring from 10 centimeters to 30 centimeters?
2. Do p. 493 # 12 A force of 800 newtons stretches a spring 70 cm on a mechanical device for
driving fence posts. Find the work done in stretching the spring the required 70 centimeters.
Homework Problems from Textbook
In exercises 1-4, determine the work done by a constant force.
1. A 100-lb bag of sugar is lifted 10 feet.
2. An electric hoist lifts a 2800-pound car 4 feet.
3. A force of 112 newtons is required to slide a cement block 4 meters in a
construction project.
4. The locomotive of a freight train pulls its cars with a constant force of 9 tons
a distance of one-half mile.
In Exercises 9-11 use Hooke’s Law to determine the variable force in the spring
problem.
9. A force of 5 pounds compresses a 15-inch spring a total of 4 inches. How
much work is done in compressing the spring 7 inches.
10.
How much work is done in compressing the spring in Exercise 9 from
a length of 10 inches to a length of 6 inches?
11.
A force of 250 newtons stretches a spring 30 centimeters. How much
work is done in stretching the spring from 20 centimeters to 50 centimeters.