Work Problems - Calculus 2

The next application for definite integrals that we are going to look at deals with the concept of work from physics. Simply put, work is the energy transferred to or from an object when it is moved by a force.

The most simple case of work is work done by a constant force, or a force that is not changing. In these cases, work is pretty simple to calculate. The work will be equal to the force, F, times the displacement, D, which is how far the object moved from where it started. This formula can be used to calculate work done by a constant force, such as lifting a 20 lb box 2 feet off the ground. The 20 lb would be the force, and the displacement would be the 2 feet. This would result in 40 ft-lbs (foot-pounds) as the amount of work done by that action if we multiply the force by the displacement.

The unit of foot-pounds is the most common unit of work used when working with the U.S. customary unit system, also known as the imperial system, but if you are working in the metric system, then the most common unit will be Joules (J), which is the same as Newton-meters. More on that in the lesson video.

Work can seem pretty simple to calculate when the force applied to an object is constant. However, not all forces are constant. Many forces will vary based on certain factors. In these instances, in order to calculate work done, we will need to used calculus, or more specifically, a definite integral.

There are various scenarios where work done by a variable force can be calculated, but the most common scenarios and types of problems encountered in Calculus 2, are spring problems, propulsion problems, lifting problems, and pumping problems.

Spring problems involve calculating the work done by stretching or compressing a spring. Propulsion problems deal with finding the work done in propelling a body from the surface of a large mass, such as Earth or the moon. In lifting problems, we want to find the work done in lifting an object where the force on that object, or the weight, varies throughout the lifting process. And finally, pumping problems involve calculating the work done in pumping a fluid, such as water, in or out of a tank.

Each of these types of problems will be covered in detail throughout the lesson and subsequent examples video on work problems. There's a lot to cover here, so let's get started!

In this lesson, you will learn:

• What work is, the appropriate units for work, and how to calculate work for scenarios with a constant force and a variable force
• The different types of work problems that can be solved with calculus, including spring, propulsion, lifting, and pumping problems
• How to identify each type of work problem, and how to set up a definite integral to calculate the work in each scenario

I hope you find this video to be helpful!

-Josh