1.3 The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus (FTC) connects differentiation and integration, showing they're inverse operations. It proves that finding the area under a curve and calculating rates of change are two sides of the same coin, which transforms how we evaluate integrals.

The theorem has two parts. The first shows that integration produces antiderivatives, while the second gives you a way to evaluate definite integrals without Riemann sums. Together, they replace tedious limit-based calculations with straightforward antiderivative evaluation.

The Fundamental Theorem of Calculus

Mean Value Theorem for Integrals

Before diving into the FTC itself, there's a supporting result you should know. The Mean Value Theorem for Integrals says that for a continuous function $f$ on $[a, b]$, there exists some point $c$ in $[a, b]$ such that:

$\int_a^b f(x)\, dx = f(c)(b - a)$

Geometrically, this means the area under the curve $f(x)$ over $[a, b]$ equals the area of a rectangle with base $b - a$ and height $f(c)$. That height $f(c)$ is the average value of $f$ on the interval.

• The theorem guarantees that a continuous function actually achieves its average value at some point in the interval.
• It relies on the Intermediate Value Theorem to ensure that point $c$ exists.
• The average value of $f$ on $[a, b]$ is given by $f_{\text{avg}} = \frac{1}{b-a}\int_a^b f(x)\, dx$.

Fundamental Theorem of Calculus

Part 1 (Differentiation of an Integral): If $f$ is continuous on $[a, b]$, then the function defined by

$F(x) = \int_a^x f(t)\, dt$

is continuous on $[a, b]$, differentiable on $(a, b)$, and satisfies $F'(x) = f(x)$.

In plain terms: if you accumulate area under a continuous function and then take the derivative of that accumulation, you get back the original function. This guarantees that every continuous function has an antiderivative.

Part 2 (Evaluation of 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 sometimes called the Newton-Leibniz formula. It means you can evaluate a definite integral by finding any antiderivative and plugging in the endpoints. No limits of Riemann sums required.

The two parts together show that differentiation and integration undo each other:

• Part 1: The derivative of the integral of $f$ gives back $f$.
• Part 2: The definite integral of $f'$ over $[a, b]$ gives the net change $f(b) - f(a)$.

Derivatives of Integrals

Part 1 of the FTC directly tells you how to differentiate integrals that have variable limits.

Variable upper limit (the standard case):

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

where $a$ is a constant and $f$ is continuous.

Variable lower limit:

$\frac{d}{dx} \int_x^b f(t)\, dt = -f(x)$

The negative sign appears because increasing $x$ shrinks the interval of integration from the left.

Variable limit with a composite function (chain rule): If the upper limit is a function $g(x)$ rather than just $x$, apply the chain rule:

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

For example, to find $\frac{d}{dx} \int_1^{x^2} \sin(t)\, dt$:

1. Identify $f(t) = \sin(t)$ and $g(x) = x^2$.
2. Apply the formula: $f(g(x)) \cdot g'(x) = \sin(x^2) \cdot 2x$.

If both limits are functions of $x$, split the integral at a constant and handle each piece separately.

Computing Definite Integrals

To evaluate $\int_a^b f(x)\, dx$ using FTC Part 2:

1. Find an antiderivative $F(x)$ of $f(x)$.
2. Evaluate $F(b) - F(a)$. This difference is often written using the bracket notation: $F(x)\Big|_a^b$.

Example: Evaluate $\int_1^3 2x\, dx$.

1. An antiderivative of $2x$ is $F(x) = x^2$.
2. Compute $F(3) - F(1) = 9 - 1 = 8$.

This is far more efficient than setting up Riemann sums, and it works for computing areas, volumes, work, and any other accumulated quantity.

Differentiation vs. Integration

These two operations are inverses, but they answer different questions:

• Differentiation finds the instantaneous rate of change (slope) of a function at a point.
• Integration finds the accumulated total (area under the curve) over an interval.

The FTC ties them together:

• FTC Part 1 says differentiating an accumulation function recovers the original rate: $\frac{d}{dx}\int_a^x f(t)\,dt = f(x)$.
• FTC Part 2 says integrating a rate of change over an interval gives the net change: $\int_a^b f'(x)\,dx = f(b) - f(a)$.

This relationship is the foundation for nearly everything else in Calculus II, from computing areas between curves to solving differential equations.

Continuity and the Fundamental Theorem

Continuity of $f$ is a key requirement for both parts of the FTC.

• Part 1 requires $f$ to be continuous so that the accumulation function $F(x) = \int_a^x f(t)\,dt$ is guaranteed to be differentiable.
• Part 2 requires $f$ to be continuous on $[a, b]$ so that an antiderivative exists and the evaluation formula holds.

If $f$ has a discontinuity on $[a, b]$, you can't apply the FTC directly. Instead, you'd need to split the integral at the discontinuity and handle each piece separately (this comes up later with improper integrals).

Also keep in mind: when the limit of integration is a composite function like $g(x)$, the chain rule becomes essential. The FTC alone gives you $f(g(x))$, but you must multiply by $g'(x)$ to account for the rate at which the upper limit itself is changing.