Learning Outcomes

Upon completing this lesson, you will be able to:

• Define an antiderivative and understand its relationship to the derivative.
• Use the notation for indefinite integrals and recognize the importance of the constant of integration, $C$.
• Understand and interpret sigma notation to represent sums.
• Explain the concept of the definite integral as the net area under a curve.
• Approximate the area under a curve using a finite number of rectangles.

Topics Covered

• Antiderivatives & Indefinite Integrals
• Sigma and Sum Notation
• The Definite Integral as an Area

I. Antiderivatives & Indefinite Integrals

The second major branch of calculus is integral calculus, which is centered around the concept of the integral. At its most basic level, integration can be thought of as the reverse process of differentiation.

A. The Antiderivative

If differentiation is the process of finding the rate of change of a function, then antidifferentiation is the process of finding a function when its rate of change is known.

A function $F$ is called an antiderivative of a function $f$ if the derivative of $F$ is $f$. That is, if

$$ F'(x) = f(x) $$

Example 1

What is an antiderivative of

$$ f(x) = 2x $$

We are looking for a function $F(x)$ such that $F'(x) = 2x$. From our knowledge of the Power Rule, we know that the derivative of $x^2$ is $2x$. So, one possible antiderivative is

$$ F(x) = x^2 $$

B. The Constant of Integration, C

Is $F(x) = x^2$ the only antiderivative of $f(x) = 2x$?

Consider the function $G(x) = x^2 + 5$. Its derivative is $G'(x) = 2x + 0 = 2x$. Similarly, the derivative of $H(x) = x^2 - 100$is $H'(x) = 2x$.

It turns out that any function of the form $F(x) = x^2 + C$, where $C$ is any constant, is an antiderivative of $f(x) = 2x$.

This is because the derivative of any constant is zero. This constant $C$ is called the constant of integration.

C. The Indefinite Integral

The family of all antiderivatives of a function $f(x)$ is called the indefinite integral of $f(x)$ with respect to $x$, and it is denoted by: