5.3 The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus bridges the gap between derivatives and integrals by showing that these two operations are inverses of each other. Instead of computing areas under curves through tedious Riemann sums, this theorem lets you evaluate definite integrals using antiderivatives. It's the single most important result in Calculus I.

The Fundamental Theorem of Calculus

Mean Value Theorem for Integrals

If $f$ is continuous on $[a, b]$, then there exists some point $c$ in $(a, b)$ such that:

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

Think about what this says geometrically: you can always find a rectangle with base $b - a$ and height $f(c)$ that has exactly the same area as the region under the curve from $a$ to $b$.

• The value $f(c)$ is the average value of $f$ on $[a, b]$. You can rearrange the formula to see this: $f(c) = \frac{1}{b-a}\int_{a}^{b} f(x)\, dx$
• Continuity of $f$ is what guarantees such a $c$ exists (by the Intermediate Value Theorem, the function must hit its average value somewhere on the interval)
• This result is used in proving Part 1 of the Fundamental Theorem

Fundamental Theorem of Calculus, Part 1

Part 1 tells you what happens when you differentiate an integral. If $f$ is continuous on $[a, b]$ and you define:

$F(x) = \int_{a}^{x} f(t)\, dt$

then $F$ is an antiderivative of $f$, which means:

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

The function $F(x)$ represents the accumulated area under $f(t)$ from $a$ to $x$. As you move $x$ to the right, the rate at which that area grows equals the height of the curve at $x$. That's why the derivative of the integral gives you back the original function.

Example: If $F(x) = \int_{1}^{x} t^2\, dt$, then $F'(x) = x^2$. You just plug the upper limit into the integrand.

Watch out for chain rule situations. If the upper limit is a function of $x$ rather than just $x$ itself, you need to apply the chain rule:

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

For instance, $\frac{d}{dx} \int_{0}^{x^2} \sin(t)\, dt = \sin(x^2) \cdot 2x$.

Fundamental Theorem of Calculus, Part 2

Part 2 gives you a practical way to evaluate definite integrals. If $f$ is continuous on $[a, b]$ and $F$ is any antiderivative of $f$, then:

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

This is the result that replaces Riemann sums with a simple subtraction. Here's the process:

1. Find an antiderivative $F(x)$ of the integrand $f(x)$
2. Evaluate $F$ at the upper limit $b$ and the lower limit $a$
3. Subtract: $F(b) - F(a)$

Example: Evaluate $\int_{0}^{1} x^2\, dx$.

1. An antiderivative of $x^2$ is $\frac{1}{3}x^3$
2. Evaluate: $F(1) = \frac{1}{3}(1)^3 = \frac{1}{3}$ and $F(0) = \frac{1}{3}(0)^3 = 0$
3. Subtract: $\frac{1}{3} - 0 = \frac{1}{3}$

Notice you don't need the $+C$ here. Any constant you add cancels out in the subtraction $F(b) - F(a)$.

Connections and Applications of the Fundamental Theorem of Calculus

Differentiation vs. integration relationship

The two parts together establish that differentiation and integration are inverse operations:

• Part 1 says: if you integrate a function and then differentiate, you get the original function back
• Part 2 says: if you find an antiderivative (undo a derivative) and evaluate at the endpoints, you get the definite integral

This inverse relationship is what makes the theorem so useful. It means you can find areas under curves without limits of Riemann sums, and you can recover total change from a rate of change by integrating.

Combined Fundamental Theorem concepts

Some problems require both parts working together.

Example: Find the area under $y = \frac{d}{dx} \int_{0}^{x} \sin(t)\, dt$ from $x = 0$ to $x = \pi$.

1. Apply Part 1 to simplify the function: $\frac{d}{dx} \int_{0}^{x} \sin(t)\, dt = \sin(x)$
2. Apply Part 2 to evaluate the area: $\int_{0}^{\pi} \sin(x)\, dx = [-\cos(x)]_{0}^{\pi} = -\cos(\pi) - (-\cos(0)) = 1 + 1 = 2$

Connections in mathematical analysis

The Fundamental Theorem provides the foundation for many topics you'll encounter later:

• Differential equations: Solving equations that relate a function to its derivatives often relies on integrating both sides
• Probability: Computing probabilities for continuous random variables requires evaluating integrals of density functions
• Multivariable and vector calculus: Theorems like Green's Theorem and Stokes' Theorem generalize the same core idea (relating a quantity on a boundary to a quantity over a region) to higher dimensions