Area Under Graphs: Finding Work Done

[IQScience]

Okay, so I've got a question on graphs and areas under curves.

For example, a force-distance graph, with force on the y-axis and distance on the x-axis, where you find work done from that graph.
I understand that a constant force will produce a horizontal line on the graph, like so:

Force
|
|-------------
|
|
|___________distance

So the area under the line is just force x distance.
This matches the equation, work = force x distance.

But I'm having trouble understanding a proportional graph, or one that looks like a triangle, like this:

Force
|------/-----
|-----/-----
|----/------
|---/-------
|--/--------
|-/---------
|/_________distance

i.e. force increases at a constant rate.

To find work done from this, you find the area under the line, which is a triangle.
But this doesn't match the equation (work = force x distance) which is confusing me.
Why is it not force x distance like the equation?
Why is is 1/2 force x distance now?
I understand that this is because the area under the line is a triangle, but why is work done the area under the graph?

 

[physwizard]

You know that work done is $\int{Fdx}$ in one dimension and you know that integral is equal to the area under the curve so work done would be equal to the area under the curve.

 

[SteamKing]

What is the average force under the second diagram?

Remember, work = force * distance applies regardless of the amount of force or the distance over which it acts. If you understand the concept of integration, you will see that to determine the work under a force-distance diagram, you are calculating incrementally all of the little bits of work done by a constant force acting over a small distance. In the limit, as these small increments of distance go to zero, then the total work is found.

 

[IQScience]

Ahhhhh, I see, that makes sense.

Thank you very much