Why do you calculate the area UNDER a curve with integration?

In summary, the conversation discussed the impracticality of finding the area outside of a closed figure and how integration of a bounded function is defined by dividing the segment into smaller segments and constructing rectangles whose area approaches the integral. It was also mentioned that the integral represents the area under the line.
  • #1
user111_23
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0
I'm think it's because it's impractical to find the area outside a closed figure. But I'm still not sure.
 
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  • #2
Because that's how it's defined? The area outside of a closed figure is going to be infinite unless you have a very contrived scenario
 
  • #3
For the same reason it doesn't make sense to find the area outside of a square, circle, or anything else.
 
  • #4
One of the ways that integration of a bounded function f on the segment [a,b] defined is by dividing [a,b] into an increasingly large number of segments. Then for each segment construct a rectangle whose width is the length of the segment and whose height is the value of the function at some point in the segment. The sum of the area of these rectangles approaches the integral as the segments get smaller.
It's easy to see that by this definition the integral is equal to the area between the function and the x-axis, or, the area "under" the line.
 

1. Why is integration used to calculate the area under a curve?

Integration is used to calculate the area under a curve because it allows us to find the exact value of the area, which would not be possible using other methods such as approximations or numerical methods. Integration also takes into account the entire curve, rather than just a few points, providing a more accurate result.

2. How is integration related to finding the area under a curve?

Integration is related to finding the area under a curve because it involves breaking down the curve into smaller, infinitesimal parts and adding them up to find the total area. This process is known as Riemann summation and is the basis of integration.

3. Can integration be used for any type of curve?

Yes, integration can be used for any type of curve, as long as it is continuous. This includes polynomial, trigonometric, exponential, and logarithmic curves, among others. However, the method of integration used may vary depending on the type of curve.

4. Why is the area under a curve important?

The area under a curve is important because it represents the total accumulation of a quantity over a given interval. This can have real-world applications in various fields, such as physics, economics, and engineering. Integration allows us to find this area accurately, making it an essential tool in scientific and mathematical calculations.

5. Is there a difference between finding the area under a curve and finding the integral of a function?

No, there is no difference between finding the area under a curve and finding the integral of a function. Integration is the mathematical process used to calculate both the area under a curve and the integral of a function. In fact, the integral of a function represents the area under the curve of that function.

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