Section 6.4
Another Application of
Integration
Definition: Work
• Work generally refers to the amount of
effort required to perform a task
More precisely…
If an object is moved a distance d in the
direction of an applied force F, the work
done by the force is
W=Fd
• A force
Examples?
pushing or pulling an object
• The downward pull of gravity on an object
More details!
• If the object moves along a straight line
with position s(t) then the force F acting on
the object in the same direction is defined
by Newton’s second law:
F = (mass)(acceleration) = ms’’(t)
Remarks: F=ms’’(t)
•
•
•
•
Mass has units in kilograms
Distance has units in meters
Time has units in seconds
F has units in (kg)(m)/s2 = N (Newton)
– In the US, Force may use units of weight
(pounds)
• W=Fd gives units of Newton-meters or
Joules
Example
• How much work is done in lifting a 1.2 kg
book off the floor to put it on a desk that is
7 m high? (assume g = 9.8 m/s2)
What if the force is not constant?
• Suppose an object moves along a straight line
from x = a to x = b by a varying force f(x).
• Partition [a,b] into subintervals of length x
*
x
• Choose a sample point i  xi 1 , xi 
• Since f(x) is a varying force and we’ll assume
that x is “small,” we can say that f(x) is almost
constant over xi 1 , xi 
• So the force acting on the object over xi 1 , xi 
is approximately f ( xi* )
Work!
• So the work done to move the particle from
xi 1 to xi is Wi  f ( x )x
*
i
• And so the total work is
n
Wi   f ( x )x
i 1
*
i
Reimann Sum!
b
W   f ( x)dx
a
Example
• When a particle is a distance x from the origin, a
force of f ( x)  x 2  2 x pounds acts on it. How
much work is done to move the object from x=1
to x=3?
A more exciting example:
Work required to move a liquid
• Suppose a tank is shaped like an inverted
circular cone with a radius of 4 meters at
the top and a height of 10 meters.
• The tank is filled to a height of 8 meters.
• Find the work required to empty the tank
by pumping the water out the top.
• Use the fact that the density of water is
1000 kg/m3
A drawing almost always helps
4
10
8
Another Example
• A spherical tank with a radius of 8 ft is half
full of a liquid that weighs 50 pounds/ft3.
• Find the work required to pump the liquid
out of a hole in the top of the tank.