Philosophical question about the integral expression

In summary, the integral expression finds the area under a curve by taking the limit as dx approaches zero, which eliminates the need to calculate an infinite number of rectangles. This concept is similar to Zeno's paradox of Achilles and the tortoise.
  • #1
Jacobim
28
0
How is it possible...that the integral performs an infinite amount of calculations to give the area under a curve.

The integral expression has to find an infinite amount of areas of the (dx by f(x)) rectangles.

I'm guessing there is a simple answer to this, I'm just not quite piecing it together.

I have probably done several homework assignments covering exactly what I'm asking, but I do not know how to answer this question to myself.
 
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  • #2
"We take the limit as dx approaches zero"

So this limit means there is no need to calculate the area of the infinite amount of rectangles. Is that it? I was hoping for something more exciting!
 
  • #3
Well, its a little more complicated that just "dx approaches 0" because the number of terms is increasing at the same time, but yes, we do NOT actually calculate and infinite number of things.
 
  • #4
It's like asking how you can sum an infinite series a + ar + ar2 + ar3 + ... where |r|< 1 to get exactly a/(1-r) without needing an infinite number of calculations.
 
  • #5
Jacobim said:
"We take the limit as dx approaches zero"

So this limit means there is no need to calculate the area of the infinite amount of rectangles. Is that it? I was hoping for something more exciting!

No, you have it. If you can compute limits without summing an infinite number of things then you can get the answer without doing an infinite amount of work. It's like Zeno's argument that Achilles can't overtake the tortoise. But only vaguely.
 

1. What is an integral expression?

An integral expression is a mathematical representation of the area under a curve. It is used to calculate the total value of a continuously changing quantity.

2. What is the purpose of using an integral expression?

The purpose of using an integral expression is to solve problems involving rates of change, such as finding the area under a curve or the total distance traveled by an object with varying velocity.

3. How is an integral expression related to calculus?

Integral expressions are a key concept in calculus. They are used to find the antiderivative of a function, which is then used to calculate the area under a curve. This is known as the fundamental theorem of calculus.

4. Can an integral expression have negative values?

Yes, an integral expression can have negative values. This can occur if the function being integrated has negative values or if the limits of integration are set in a way that results in a negative area under the curve.

5. How do you evaluate an integral expression?

There are various methods for evaluating an integral expression, such as using integration by parts, substitution, or using a table of common integrals. The process involves finding the antiderivative of the function and then substituting the limits of integration into the expression to calculate the final value.

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