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 (2*x*_{0}, 2*x*_{0}).

*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=2x*_{0}.

*Step V: *Evaluate the integral.