Finding work done in compressing (forging) a solid cylinder

[MechEngrStdnt]

Homework Statement

The problem is asking to find the change in temperature of forged cylinder. I know how to calculate this,

, however I am struggling to find the work done on the cylinder, which is required to solve this problem. The cylinder is annealed copper, which is compressed from a height on 2in to 0.6in. I was able to find the forging forces (attached below in excel screen shot), and they are correct. I am given the density: 0.37336lb/in^3, and the specific heat: 861in-lb/lb°F, now I just need to find the work done on the cylinder.

Homework Equations

The Attempt at a Solution

When the coefficient of friction is 0, the work is easy to find,

However once the coefficient of kinetic friction is non-zero, this equation no longer applies.

I tried finding the area under the curve by plotting in excel and extracting the equation, however the values I am getting are not even close, I'm pretty sure I'm making some conceptual mistake, any help is much appreciated.

 

[MechEngrStdnt]

Figured it out, bounds were incorrect. X axis was in terms of percentage increase, not actual increase.