Fundamental Theorem of Calculus

Why This Matters

The Fundamental Theorem of Calculus (FTC) is arguably the most important result in all of calculus—it's the bridge that connects the two major operations you've been learning: differentiation and integration. You're being tested not just on whether you can apply the theorem mechanically, but on whether you understand why these two seemingly different processes are actually inverse operations. Exam questions will probe your ability to evaluate definite integrals using antiderivatives, interpret accumulation functions, and recognize when each part of the theorem applies.

This theorem unlocks everything from calculating areas under curves to modeling real-world accumulation problems in physics, economics, and biology. The key concepts you need to master include the evaluation of definite integrals, the derivative of integral functions, accumulation functions, and the inverse relationship between differentiation and integration. Don't just memorize the formulas—know what each part of the FTC tells you about how functions behave and why the connection between derivatives and integrals exists in the first place.

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The Two Parts of the FTC

The Fundamental Theorem comes in two parts, and understanding which part to use—and when—is critical for exam success. Each part reveals a different direction of the derivative-integral relationship.

Part 1: The Evaluation Theorem

• Converts definite integrals into antiderivative evaluation—if $f$ is continuous on $[a, b]$ and $F$ is any antiderivative of $f$, then $\int_a^b f(x) \, dx = F(b) - F(a)$
• Continuity is required for the theorem to hold; discontinuities on the interval invalidate direct application
• Transforms infinite Riemann sum calculations into simple subtraction, making area calculations practical and efficient

Part 2: The Derivative of an Integral Function

• Defines how differentiation undoes integration—if $F(x) = \int_a^x f(t) \, dt$, then $F'(x) = f(x)$ wherever $f$ is continuous
• The variable of integration ($t$) is a dummy variable; the upper limit ($x$) is what matters for differentiation
• Chain rule extension applies when the upper limit is a function: if $F(x) = \int_a^{g(x)} f(t) \, dt$, then $F'(x) = f(g(x)) \cdot g'(x)$

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The Inverse Relationship

The FTC reveals that differentiation and integration are fundamentally inverse operations—understanding this conceptually is just as important as applying the formulas. This inverse relationship is the theoretical backbone of calculus.

Differentiation as Rate of Change

• Measures instantaneous rate of change—the derivative $f'(x)$ tells you how fast $f(x)$ is changing at any point
• Local behavior focus means derivatives describe what's happening at a single instant or location
• Undone by integration when you integrate a rate of change, you recover the original quantity (plus a constant)

Integration as Accumulation

• Accumulates quantities over intervals—the integral $\int_a^b f(x) \, dx$ adds up infinitely many infinitesimal contributions
• Global behavior focus means integrals capture total change or total area across an entire interval
• Undone by differentiation when you differentiate an accumulation function, you recover the rate being accumulated

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Antiderivatives and Indefinite Integrals

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

The Antiderivative Concept

• Definition: $F$ is an antiderivative of $f$ if $F'(x) = f(x)$ for all $x$ in the domain
• Antiderivatives are not unique—if $F(x)$ is an antiderivative, so is $F(x) + C$ for any constant $C$
• The "+ C" matters for indefinite integrals but cancels out in definite integral evaluation since $(F(b) + C) - (F(a) + C) = F(b) - F(a)$

Indefinite Integral Notation

• Represents a family of functions—written as $\int f(x) \, dx = F(x) + C$, where $C$ is the constant of integration
• No bounds means no single answer—indefinite integrals give you all possible antiderivatives
• Essential for FTC Part 1 because you must first find an antiderivative before evaluating at the bounds

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Accumulation Functions

Accumulation functions are integral functions with a variable upper limit—they're central to Part 2 of the FTC and appear frequently in AP exam questions. These functions model how a quantity builds up as you move along an interval.

Defining Accumulation Functions

• Standard form is $F(x) = \int_a^x f(t) \, dt$—this represents the accumulated "area" from $a$ to $x$
• Output is net signed area under $f(t)$ from $a$ to $x$; areas below the x-axis count as negative
• Starting value $F(a) = 0$ always, since integrating from $a$ to $a$ gives zero area

Graphical Interpretation

• $F(x)$ increases where $f(x) > 0$—positive values of $f$ add to the accumulated total
• $F(x)$ decreases where $f(x) < 0$—negative values of $f$ subtract from the accumulated total
• $F$ has extrema where $f$ crosses zero—since $F'(x) = f(x)$, zeros of $f$ are critical points of $F$

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Applications and Problem-Solving

The FTC isn't just theoretical—it's your primary tool for solving practical problems involving areas, accumulated quantities, and rates of change. Recognizing which application fits the problem is key to exam success.

Area Under Curves

• Computes exact areas between a function and the x-axis over $[a, b]$ using $\int_a^b f(x) \, dx = F(b) - F(a)$
• Net vs. total area distinction—the integral gives net signed area; for total area, integrate $|f(x)|$
• Geometric interpretation connects algebraic calculation to visual understanding of the curve's behavior

Real-World Modeling

• Physics applications include displacement from velocity ($\int v(t) \, dt$) and work from force
• Economics applications include total cost from marginal cost and consumer/producer surplus calculations
• Biology applications include population growth from growth rates and 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 and contrast: 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?