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

In summary, if you have a physics exam where you need to estimate the area under a curve to find the work done, the best method would be to divide the area into several trapezoids. This is more accurate than using rectangular bars and can be easily calculated. However, if you only have a graphical representation of the curve, you can also approximate it by drawing a sloping straight line that has the same area as the curve. This method is quick and surprisingly accurate, but may require using multiple straight lines for a highly curved curve. Overall, it is important to consider the shape of the curve and choose the best method accordingly.
  • #1
slingboi
8
0
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?
 
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  • #2
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
 
  • #3
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.
 

1. How can I estimate the area underneath a curve?

One way to estimate the area underneath a curve is by using the trapezoidal rule, which involves dividing the curve into smaller trapezoids and calculating the area of each one.

2. Is there a quicker method to estimate the area under a curve?

Another quick method to estimate the area under a curve is by using the midpoint rule, which involves dividing the curve into smaller rectangles and using the midpoint of each rectangle to calculate the area.

3. Can I use software to estimate the area under a curve?

Yes, there are various software programs, such as Microsoft Excel or Wolfram Alpha, that have built-in functions for estimating the area under a curve. These programs use numerical integration methods to quickly and accurately calculate the area.

4. Do I need to know the equation of the curve to estimate the area underneath it?

No, you do not need to know the exact equation of the curve to estimate its area. Both the trapezoidal and midpoint rules only require the values of the curve at specific points, which can be obtained through data or by plotting the curve on a graph.

5. Are there any limitations to estimating the area under a curve using numerical methods?

Yes, there are limitations to estimating the area under a curve using numerical methods. These methods may not provide an exact value for the area and are subject to errors. Additionally, they may not work well for curves with complex shapes or rapidly changing slopes.

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