Welcome back, class! Today, we arrive at the most pivotal moment in our course: The Fundamental Theorem of Calculus (Section 5-3). As the name suggests, this theorem is the bedrock of everything we do in higher-level mathematics. It connects the two main branches of calculus—differential calculus (the problem of tangents) and integral calculus (the problem of areas)—into one unified framework.

The Big Idea

Until now, we have treated derivatives and integrals as separate topics. The Fundamental Theorem of Calculus (FTC) reveals that differentiation and integration are actually inverse processes. Roughly speaking, integration "undoes" differentiation, and vice versa.

We generally break this theorem down into two parts. Let's look at what you need to know from the lecture notes and the PowerPoint.

Part 1: The Evaluation Theorem

This is the part of the theorem you will use most frequently for calculation. It states that if $f$ is continuous on the interval $[a, b]$ and $F$ is any antiderivative of $f$, then:

$$\int_{a}^{b} f(x) \, dx = F(b) - F(a)$$

Why is this exciting? It means we no longer need to use tedious Riemann Sums and limits to calculate the area under a curve! We simply need to:

  • Find the antiderivative $F(x)$.
  • Plug in the upper bound ($b$).
  • Plug in the lower bound ($a$).
  • Subtract the two results.

Part 2: The Derivative of an Integral

The second part of the theorem focuses on the accumulation function. It tells us that if we define a function $g(x)$ as an integral with a variable upper bound, the derivative of that function returns us to the original integrand.

$$\frac{d}{dx} \left[ \int_{a}^{x} f(t) \, dt \right] = f(x)$$

Professor Baker's Tip: Be careful when the upper limit is not just $x$, but a function of $x$ (like $x^2$ or $\sin x$). In these cases, you must apply the Chain Rule:

$$\frac{d}{dx} \left[ \int_{a}^{g(x)} f(t) \, dt \right] = f(g(x)) \cdot g'(x)$$

Key Takeaways for Section 5-3

As you review the attached PowerPoint and video lecture, focus on these skills:

  • Notation matters: When evaluating $\int_a^b$, we often use the bracket notation $[F(x)]_a^b$ to show our work before subtracting.
  • Constants cancel out: Note that when finding the definite integral, we don't need to write $+ C$, because $(F(b) + C) - (F(a) + C) = F(b) - F(a)$. The constants cancel!
  • Continuity is key: Remember that the theorem only guarantees results if the function is continuous over the interval.

Don't be intimidated by the notation. Once you practice a few examples, you will see the beautiful symmetry in how these operations interact. Be sure to download the notes below and follow along with the examples in the video.

Happy Integrating!