This might be a trivial question but is illustrated below.

Why is the area 'below' the graph always taken for a velocity-time graph when finding the displacement? I mean why is the area with the time axis taken? Why not with the other axis? I know it sounds silly but what is the definitive answer to this question? What is the link with the variable with respect to which we want to perform integration?

For an elastic material the energy stored is calculated from a Force-extension graph by considering the area with the displacement axis. Why always this?

For a graph with straight line through origin, the area with both axes is equal. My problem comes when the graph is a curve so that the areas are not equal.

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    $\begingroup$ Where is the illustration? $\endgroup$ – Dave Coffman Apr 4 '15 at 18:36

It follows from the definitions. Velocity is defined as the time-derivative of displacement, $$ v=\frac{dx}{dt} $$ from which we get $$ x=\int v(t)\,dt $$ which you may recall is the area under the $v(t)$ graph.

For the elastic energy case, start from the definition of force, $f=-dU/dx$, and follow the same argument.

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