A particle moves along a line at 45 to the x-axis as shown in the diagram. A force acts on the particle, given by

Find the work done on the particle as it moves from (0,0) to (2x0, 2x0).

Step I: Choose a coordinate system and model the system as a collection of differential elements:

 

The coordinate system is already given. We model the path as a collection of differential steps

 

Step II: Identify a typical element. Draw it on the diagram. Describe it using the coordinates.

 

A typical element is:

at coordinates x,y where y=x.

 

Step III: Express the contribution of this element to the desired sum in terms of the coordinates.

The contribution is the work

Step IV: Determine the limits of integration.

The limits are x=0 to x=2x0.

 

Step V: Evaluate the integral.