Please could someone explain to me WHY calculating a definite integral finds you the area under a curve?
Integrating to Find Area Under a Curve Watch
- Thread Starter
- 29-02-2012 19:57
- 29-02-2012 20:01
Wikipedia does a pretty good job of giving a rough idea why it's the case, check out the 'Geometric intuition' bit of http://en.wikipedia.org/wiki/Fundame...em_of_calculus
- 29-02-2012 20:02
The answer to this question is a lot easier to understand graphically, but basically you have to imagine that in , the f(x) is the height of a rectangle and the dx is some arbitrarily small width of that rectangle. The integral means, in essence, adding up all the rectangles - which amounts to finding the total area under the curve.
Edit: this is not really super helpful in terms of answering your question, but is roughly relevant and amused me. Why don't I think of things like this?
Last edited by teamnoether; 29-02-2012 at 20:21.
- 29-02-2012 20:03
tbh, it depends what you define an integral to be.
- 29-02-2012 20:05
You would have probably studied calculating area by using trapezium rule, in that rule we use strips.
In integration, we use infinite strips.
The area of one strip will be, Area= where delta x is very small, almost zero.
Now summing the strips,
As , the summation becomes integration,
Hope i am correct.
- 01-03-2012 00:22
Strictly speaking, it's because the integral is what we define the "area under a curve" to be, and the fact it's consistent with the definition of area for simple shapes is what allows this definition to make sense.Last edited by matt2k8; 01-03-2012 at 00:27.
- 01-03-2012 00:44
the area (called the Riemann integral) is the limit of the riemann sum over the certain interval - end points (a,b), divided into intervals of equal length b-a/l, with a point c in each interval. Sum is then SIGMA i=0 to n-1 of f(c_i)(x_i+1-x_i)