# Difference between integral and a function

• B
• kolleamm
In summary: Online it says that the integral is the opposite of the derivative. So x^2 is the integral of 2x.No, the integral of 2x is x^2 + c. They are not exactly symmetric.Both x2 and 2x are functions, and they are different functions. The two are related through integration and differentiation, as explained by the first fundamental theorem of calculus.To be clear, we should label the two functions differently:F(x) = x2f(x) = 2xF(x) and f(x) are different functions, with f(x) being the derivative of F(x). Stated in math terminology, F'
kolleamm
Online it says that the integral is the opposite of the derivative. So x^2 is the integral of 2x.

So if f(x) = x^2 , does that mean that the integral is just the function itself? Basically whatever f(x) equals?

kolleamm said:
Online it says that the integral is the opposite of the derivative. So x^2 is the integral of 2x.
No, the integral of 2x is x^2 + c. They are not exactly symmetric.
So if f(x) = x^2 , does that mean that the integral is just the function itself? Basically whatever f(x) equals?
No, for the reason just stated.

Both x2 and 2x are functions, and they are different functions. The two are related through integration and differentiation, as explained by the first fundamental theorem of calculus.

To be clear, we should label the two functions differently:

F(x) = x2
f(x) = 2x

F(x) and f(x) are different functions, with f(x) being the derivative of F(x). Stated in math terminology, F'(x) = f(x) and ∫f(x)dx = F(x).

kolleamm said:
Online it says that the integral is the opposite of the derivative. So x^2 is the integral of 2x.

So if f(x) = x^2 , does that mean that the integral is just the function itself? Basically whatever f(x) equals?

You're question is a bit confusing. So let's start with a smooth function ##f: \mathbb{R} \rightarrow \mathbb{R}##, e.g. ##f(x)=x^2##.
(I picked smoothness here for reason of simplicity and it means mathematically basically the same as in ordinary language: no gaps or corners, pretty smooth.)

Then the derivative ##f\,' ## of ##f## is again a smooth function that associates to each point ##x## the slope of ##f## which is defined as the steepness of its tangent, i.e. the steepness of the touching line. In the example we have ##f\,'(x)=2x## which means, that e.g. at the point ##x=5## we have a slope of ##2\cdot 5 = 10## (##10## inches of gained height on ##1## inch horizontal deviation).

The integral ##\int f(x)\, dx = F(x)## of ##f## is a function ##F## that measures the area between the ##x-##axis and the function ##f##. Basically as we measure the area of a room: length times width. Only that it also works for curved walls. In the example we have ##\int x^2 dx = \frac{1}{3}x^3## which means, that the area ##A## underneath the parabola ##x^2##, say between the points ##x=1## and ##x=4##, is ##A= \frac{1}{3}\cdot 4^3 - \frac{1}{3} 1^3 = 21##.

Now what makes a derivative and an integral sort of "opposite" is, that ##\int f\,'(x) dx = f(x)##, or in our example ##\int 2x \,dx= x^2##.

thank you for your help although I'm still confused

## 1. What is the difference between an integral and a function?

An integral is a mathematical concept that represents the accumulation of a quantity over a given interval. It is denoted by the symbol ∫ and is used to find the area under a curve. On the other hand, a function is a relation between two variables, where each input has a unique output. It is denoted by the symbol f(x) and is used to describe the relationship between the input and output values.

## 2. How are integrals and functions related?

Integrals and functions are closely related as integrals can be used to find the value of a function over a particular interval. In other words, the integral of a function represents the area under the curve of that function.

## 3. Can a function be an integral?

No, a function cannot be an integral. A function is a mathematical concept that describes a relationship between two variables, while an integral is a mathematical operation used to find the area under a curve.

## 4. How do you find the integral of a function?

To find the integral of a function, you can use integration techniques such as the Fundamental Theorem of Calculus or integration by substitution. These techniques involve finding an antiderivative of the function, which is a function whose derivative is equal to the original function.

## 5. What are the applications of integrals and functions?

Integrals and functions have various applications in mathematics and other fields such as physics, engineering, and economics. They are used to solve problems involving area, volume, and motion, among others. In real-life situations, they can be used to calculate quantities such as work, displacement, and profit.

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