What way could I estimate, easiest and quickest, the area underneath a curve?

[slingboi]

I have a physics exam soon and i would just like to know, which method would be best for me to use to estimate the area under a curve to estimate work done when the force-extension graph is not linear. I know you could draw bars to estimate the area of the curve and you could draw a line so that the area underneath the line equals an estimate of the actual curve, but is there any better way I could use in the exam because drawing bars could take a long time? What do you think would be the best method for me to use to maintain some accuracy?

 

[tommw]

What kind of exam is that? I wouldn't think that you can be expected to give an accurate estimate of an area underneath a curve without knowing anything else.

So in the case you really only have a graphical representation of the dependency f(x), the best way would indeed be to use one of the methods you described, but that depends of course on the shape of the curve.
In general deviding the area to integrate into several trapezoids is a good idea. This is more accurate than using rectangular bars and can be as easily calculated. In most cases, drawing 3 to 5 trapezoids should suffice if you don't have to deal with a really funny-shaped curve.
see https://en.wikipedia.org/wiki/Trapezoidal_rule

 

[Philip Wood]

I find that approximating the curve by a straight line (usually sloping) such that, by eye, the area under the straight line is the same as that under the curve, is quick and surprisingly accurate. You judge your straight line so that the area by which it undercuts the curve along part of its length, equals that by which it overestimates in the other part of its length. If the original curve is highly curved you might have to use one straight line for part of it, and another for the other part.