Fundamental Theorem of Calculus

Why This Matters

The Fundamental Theorem of Calculus (FTC) connects the two major operations in calculus: differentiation and integration. It shows that these two seemingly different processes are actually inverse operations. Beyond just applying the theorem mechanically, you need to understand why this connection exists and how to recognize which part of the theorem a problem is asking you to use.

The FTC is what makes it possible to evaluate definite integrals without grinding through Riemann sums. It also powers real-world applications from calculating areas under curves to modeling accumulation problems in physics and economics. The core skills here are evaluating definite integrals with antiderivatives, differentiating integral functions, interpreting accumulation functions, and understanding the inverse relationship between differentiation and integration.

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The Two Parts of the FTC

The Fundamental Theorem comes in two parts. Each part reveals a different direction of the derivative-integral relationship, and knowing which one to reach for is half the battle on any problem.

Part 1: The Evaluation Theorem

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 converts a definite integral into simple subtraction. Instead of computing an infinite Riemann sum, you just find an antiderivative, plug in the bounds, and subtract.

• Continuity is required on the interval for the theorem to apply. Discontinuities on $[a, b]$ mean you can't use this directly.
• The notation $F(x) \Big|_a^b$ is shorthand for $F(b) - F(a)$.

Part 2: The Derivative of an Integral Function

If $f$ is continuous and $F(x) = \int_a^x f(t) \, dt$, then:

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

Differentiation undoes integration. The variable $t$ inside the integral is a dummy variable; what matters is that the upper limit is $x$.

When the upper limit is a function $g(x)$ instead of just $x$, you need the chain rule:

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

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The Inverse Relationship

The FTC tells you that differentiation and integration undo each other. This is the conceptual core of the theorem, and understanding it will help you reason through problems even when the algebra gets complicated.

Differentiation as Rate of Change

• The derivative $f'(x)$ measures the instantaneous rate of change of $f(x)$ at a point.
• Derivatives describe local behavior: what's happening at a single instant.
• Integration reverses this. When you integrate a rate of change over an interval, you recover the total change in the original quantity.

Integration as Accumulation

• The integral $\int_a^b f(x) \, dx$ adds up infinitely many infinitesimal contributions over an interval.
• Integrals describe global behavior: the total change or total area across an entire interval.
• Differentiation reverses this. When you differentiate an accumulation function, you recover the rate being accumulated.

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Antiderivatives and Indefinite Integrals

Before you can use Part 1 of the FTC, you need to find antiderivatives. An antiderivative reverses differentiation.

The Antiderivative Concept

$F$ is an antiderivative of $f$ if $F'(x) = f(x)$ for all $x$ in the domain. Antiderivatives are never unique: if $F(x)$ works, then $F(x) + C$ also works for any constant $C$, because the derivative of a constant is zero.

The "+ C" matters for indefinite integrals, but it cancels out in definite integral evaluation:

$(F(b) + C) - (F(a) + C) = F(b) - F(a)$

This is why you don't need to worry about $C$ when using FTC Part 1.

Indefinite Integral Notation

The indefinite integral $\int f(x) \, dx = F(x) + C$ represents the entire family of antiderivatives of $f$. There are no bounds, so there's no single numerical answer.

• No bounds → family of functions (with $+C$)
• With bounds → a specific number (net signed area)

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Accumulation Functions

An accumulation function has a variable upper limit and is central to FTC Part 2. These functions model how a quantity builds up as you move along an interval.

Defining Accumulation Functions

The standard form is:

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

This gives the net signed area under $f(t)$ from $a$ to $x$. Areas below the x-axis count as negative. Notice that $F(a) = 0$ always, since integrating from $a$ to $a$ gives zero area.

Graphical Interpretation

Since $F'(x) = f(x)$ by FTC Part 2, the graph of $f$ tells you everything about the behavior of $F$:

• Where $f(x) > 0$, $F(x)$ is increasing (positive values add to the total).
• Where $f(x) < 0$, $F(x)$ is decreasing (negative values subtract from the total).
• Where $f$ crosses zero, $F$ has a critical point. If $f$ goes from positive to negative, $F$ has a local max. If $f$ goes from negative to positive, $F$ has a local min.

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Applications and Problem-Solving

The FTC is your primary tool for computing areas, accumulated quantities, and connecting rates to totals.

Area Under Curves

Using FTC Part 1, the exact area between $f(x)$ and the x-axis over $[a, b]$ is $\int_a^b f(x) \, dx = F(b) - F(a)$.

Watch out for the net vs. total area distinction. The definite integral gives net signed area: regions below the x-axis contribute negative values. If a problem asks for total area (or total distance traveled), you need to integrate $|f(x)|$, which means splitting the integral at the zeros of $f$ and taking absolute values.

Real-World Modeling

• Physics: Displacement from velocity ($\int v(t) \, dt$), work from force
• Economics: Total cost from marginal cost, consumer and producer surplus
• Biology: Population change from growth rates, total drug concentration over time

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Quick Reference Table

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Self-Check Questions

1. If $F(x) = \int_2^x (t^3 - 4t) \, dt$, what is $F'(x)$, and at what values of $x$ does $F$ have critical points?

2. Compare: How does evaluating $\int_0^3 x^2 \, dx$ using FTC Part 1 differ from finding $\frac{d}{dx} \int_0^x t^2 \, dt$? What does each calculation tell you?

3. An accumulation function $G(x) = \int_0^x f(t) \, dt$ is increasing on $(1, 4)$ and decreasing on $(4, 7)$. What can you conclude about $f(x)$ on these intervals?

4. Which part of the FTC would you use to find $\frac{d}{dx} \int_1^{x^2} \sin(t) \, dt$, and what additional rule must you apply?

5. Explain why the constant of integration $C$ appears in indefinite integrals but doesn't affect the value of definite integrals. How does this connect to the FTC?