A little confused about integrals

In summary, integrals are used to find the area under a curve, specifically the area under the curve of the original function. The antiderivative of a function is closely related to this area and can be used to calculate it more easily. It is defined as the function whose derivative is the original function.
  • #1
taylor__hasty
4
0
I learned that integrals are finding the area under a curve. But I seem to be a little confused. Area under the curve of the derivative of the function? Or area under the curve of the original function?

If an integral is the area under a curve, why do we even have to find the anti derivative at all?
 
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  • #2
taylor__hasty said:
I learned that integrals are finding the area under a curve. But I seem to be a little confused. Area under the curve of the derivative of the function? Or area under the curve of the original function?

If an integral is the area under a curve, why do we even have to find the anti derivative at all?

An integral is the area under the curve of the (original) function.

This area is, however, closely related to the anti-derivative of the function. This is the Fundamental Theorem of Calculus.
 
  • #3
taylor__hasty said:
I learned that integrals are finding the area under a curve. But I seem to be a little confused. Area under the curve of the derivative of the function? Or area under the curve of the original function?

If an integral is the area under a curve, why do we even have to find the anti derivative at all?

The antiderivative (if available) is useful because if ##A(a,z)## is the area under the curve ##y = f(x)## from ##x=a## to ##x=z## (with ##a## some fixed number), and if ##F(x)## is the antiderivative of ##f(x)##, then we have:
[tex] \text{area} = A(a,z) = F(z) - F(a)[/tex]
(Fundamental Theorem of Calculus).

So, if you know the antiderivative function you can use it to calculate the area. That is a lot easier than splitting up the interval ##[a,z]## into 100 billion little intervals and adding up the 100 billion little rectangular areas (which would give a good approximation to the actual area---not necessarily the exact area).
 
  • #4
taylor__hasty said:
I learned that integrals are finding the area under a curve. But I seem to be a little confused. Area under the curve of the derivative of the function? Or area under the curve of the original function?

If an integral is the area under a curve, why do we even have to find the anti derivative at all?

Integration can be used to find the area under a curve, but that is just one trivial function of the integration tool.

As to anti-derivatives: The definition is that F is the anti-derivative of f if dF/dx = f. This has one important consequence:

F is differentiable. f does not have to be. I fact, f does not even have to be continuous.
 

1. What is an integral?

An integral is a mathematical concept used to find the area under a curve or the accumulation of a changing quantity. It is the reverse process of differentiation and is represented by the symbol ∫.

2. What is the difference between definite and indefinite integrals?

A definite integral has specific limits of integration, while an indefinite integral does not. A definite integral will give a numerical value, while an indefinite integral will give an equation with a constant term.

3. How do I solve an integral?

To solve an integral, you need to use integration techniques such as substitution, integration by parts, or trigonometric substitution. It is important to also identify the limits of integration and use them in your solution.

4. What is the purpose of integrals in real life?

Integrals have many applications in real life, such as calculating the distance traveled by a moving object, finding the volume of irregularly shaped objects, and determining the total cost or profit in business.

5. How can I improve my understanding of integrals?

To improve your understanding of integrals, it is important to practice solving different types of integrals, understand the underlying concepts, and review the properties and rules of integration. It can also be helpful to seek guidance from a tutor or teacher if you are struggling with a specific concept.

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