Page 62 - Calculus for the Clueless, Calc II

Chapter 10—
Work, Work, Work

This topic is usually presented in a physics book and scares everyone to death. If done my way, I'm pretty sure
it won't bother you again. Work is defined as force times distance. However if the force is a function of
distance, theory tells us that work is the integral of F(x) dx. We give the usual examples: springs and a couple
on pumping water over the top.

Example 1—

It takes a force of 20 pounds to stretch a 7-foot spring to 11 feet. How much work to pull out the spring from 13
feet to 27 feet?

For a spring force F = kx. F = 20, and x = 11 - 7 = 4, the distance the spring is stretched from its natural length.
So k = 5 and F = 5x.

Lower limit = 13 - 7 = 6, upper limit
27 - 7 = 20.

Example 2—

How much work is done to pump the water out of a full cylindrical can radius 10 feet, height 20 feet—if the
water is to be pumped over the top?

W = density (weight/volume) x volume x height over which the water is pumped. Each section of water pumped
is a thin cylinder V = π(10) dy. Height pumped (see the figure) is 20 - y. The density for water is
2
approximately 62.5 pounds per cubic foot.

Note 1

If the outlet were 17 feet over the can, we would have 37 - y.

Note 2