Why do we integrate a function to find the area under it?

In summary: If so, then that's not really an assertion, it's just a consequence of the fundamental theorem. Left/right/middle sums all take too long to calculate the area? :wink:@Juwane: how do get your assertion from the wikipedia quote? If A_f(x) denotes the area function of some function f(x), then:wiki (i.e. the fundamental theorem) says A_f'(x)=f(x);you're saying that A_f(x)=A_{f'}(x). If so, then that's not really an assertion, it's just a consequence of the fundamental theorem.
  • #1
Juwane
87
0
Why, when finding the area by definite integral, we have to find the indefinite integral first? As I understand, to find the area of under the curve, all we need is the equation of the curve. On the other hand, the indefinite integral helps us to find the original function from its derivative. So what does this have to do with finding the area?
 
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  • #2
Juwane said:
Why, when finding the area by definite integral, we have to find the indefinite integral first?
You don't have to. If you have, in general, infinite time at your disposal. :smile:

As I understand, to find the area of under the curve, all we need is the equation of the curve. On the other hand, the indefinite integral helps us to find the original function from its derivative. So what does this have to do with finding the area?


That is the truly beautiful insight in the fundamental theorem of calculus:

To sum up the area beneath some curve, essentially an INFINITE process, can trivially be done by finding an anti-derivative to the defining curve.
 
  • #3
arildno said:
[...]

That is the truly beautiful insight in the fundamental theorem of calculus:

To sum up the area beneath some curve, essentially an INFINITE process, can trivially be done by finding an anti-derivative to the defining curve.

One of the things that the fundamental theorem of calculus tells us that the area under a curve is equal to the area under that curve's derivative, right?
 
  • #4
Juwane said:
One of the things that the fundamental theorem of calculus tells us that the area under a curve is equal to the area under that curve's derivative, right?

No, it tells us that integration (which is defined in a way that has nothing to do with derivatives or anti-derivatives) is opposite to derivation.

Which helps us in calculating integrals more efficiently.
 
  • #5
But it is true that the area under a curve is equal to the area under that curve's derivative, right?
 
  • #6
Juwane said:
But it is true that the area under a curve is equal to the area under that curve's derivative, right?

No. The area under y = 1 from x = 0 to 1 is 1. The area under the derivative is 0.
 
  • #9
Juwane: That quote you gave was just a restatement of the Fundamental Theorem of Calculus, nicksauce already gave an excellent example of where your assertion is false.
 
  • #10
arildno said:
You don't have to. If you have, in general, infinite time at your disposal. :smile:

Left/right/middle sums all take too long to calculate the area? :wink:
 
  • #11
@Juwane: how do get your assertion from the wikipedia quote? If [tex]A_f(x)[/tex] denotes the area function of some function f(x), then:

wiki (i.e. the fundamental theorem) says [tex]A_f'(x)=f(x)[/tex];
you're saying that [tex]A_f(x)=A_{f'}(x)[/tex].
 

1. Why do we need to integrate a function?

Integrating a function allows us to find the area under the curve of the function, which is useful in many real-world applications such as calculating volumes, finding work done, and determining probabilities.

2. What is the purpose of finding the area under a curve?

The area under a curve represents the accumulation of a quantity over a given interval. In other words, it gives us a measure of the total amount of something over a specific range of values.

3. How does integration help us understand a function better?

Integrating a function allows us to analyze its behavior over a given interval. It helps us identify key features such as maximum and minimum values, inflection points, and areas of rapid change.

4. Can we find the area under any type of function?

Yes, we can integrate any continuous function to find the area under it. However, the process may be more complex for certain types of functions, such as trigonometric or logarithmic functions.

5. Why do we use integration instead of simply finding the area using geometry?

Integrating a function allows us to find the area under a curve even if it does not have a simple geometric shape like a rectangle or triangle. It also allows us to find the area accurately, even for functions with infinitely many points, which is not possible with basic geometric methods.

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