Integrals
- In this section, we will introduce the concept of integrals, which is a fundamental topic of calculus
- Integrals can help us calculate the area of the region under a curve
- We will first talk about the definition of the integral, then talk more about indefinite integrals and methods of computing them
- Finally, we wil discuss definite integrals and how they differ from indefinite integrals, as well as methods of computing them
Integral Definition
Derivative Review
In the past sections, we were given a function, suppose [latex] f(x) = x^2 [/latex]. What would the derivative of this be? Well, as we know by now, the derivative of this function is 2x.
So: [latex] \frac{d}{dx}x^2 = 2x[/latex]
Integrals
Let’s take a look back at the previous example. What is the antiderivative of [latex] f(x) = 2x [/latex]? Well, let us first define antiderivatives.
Given a function f(x) the antiderivative of f(x) is any function F(x) such that the derivative of F(x) is f(x).
[latex]F'(x) = f(x) [/latex]. So, based on this definition, the antiderivative of [latex] f(x) = 2x [/latex] is [latex] F(x) = x^2 [/latex].
Finding the integral of a function refers to finding the antiderivative of the function.
Indefinite Integrals
So, we know that F(x) is the anti derivative of f(x). The indefinite integral is essentially the general form for the anti derivative of a function, and it is defined as:
[latex] \int f(x) \dx = F(x) +c[/latex]
Above, [latex]\int[/latex] is the integral symbol and f(x) is the integrand, or the function being integrated. The variable x is the integration variable and F(x) denotes the antiderivative. Finally, there is a constant c which is added known as the constant of integration.
The constant of integration represents the constant term of the original function, which could have any arbitrary value. As we know, the derivative of any constant is zero.
So, now we going back to what we were talking about before, we can say that:
[latex]\int 2xdx = x^2 + c [/latex]
Remember, the + c is added because no matter what value of c is there, let’s say: [latex] f(x) = x^2+3 or f(x) = x^2+4[/latex], the derivative of the function will still be [latex] 2x [/latex]
Computing Indefinite Integrals
CHECKPOINT!
Comfortable with derivatives? Please proceed to the next page to learn about Integrals! Otherwise, continue following along.
Definite Integrals
Integral Rules & Common Integrals
Integrals Section Complete!
Well done! In the next section, we will introduce you to the concept of integrals!
