5.5. Applications from Physics, Engineering, and Statistics http://www.ck12.org
5.5 Applications from Physics, Engineering, and Statistics
Learning Objectives
A student will be able to:
- Learn how to apply definite integrals to several applications from physics, engineering, and applied mathe-
matics such as work, fluids statics, and probability.
In this section we will show how the definite integral can be used in different applications. Some of the concepts
may sound new to the reader, but we will explain what you need to comprehend as we go along. We will take three
applications: The concepts of work from physics, fluid statics from engineering, and the normal probability from
statistics.
Work
Work in physics is defined as the product of the force and displacement. Force and displacement are vector
quantities, which means they have a direction and a magnitude. For example, we say the compressor exerts a
force of 200 Newtons(N)upward. The magnitude here is 200 N and the direction is upward. Lowering a book from
an upper shelf to a lower one by a distance of 0.5 meters away from its initial position is another example of the
vector nature of the displacement. Here, the magnitude is 0.5 m and the direction is downward, usually indicated by
a minus sign, i.e., a displacement of− 0 .5 m. The product of those two vector quantities (called theinner product,
see Chapter 10) gives the work done by the force. Mathematically, we say
W=F d,whereFis the force anddis the displacement. If the force is measured in Newtons and distance is in meters, then
work is measured in the units of energy which is in joules(J).
Example 1:
You push an empty grocery cart with a force of 44 N for a distance of 12 meters.How much work is done by you
(the force)?
Solution:
Using the formula above,
W=F d
= ( 44 )( 12 )
=528 J.Example 2: