Fundamental Theorems of Calculus

Why This Matters

The Fundamental Theorems of Calculus represent one of the most profound connections in all of mathematics—they reveal that differentiation and integration, which seem like completely different operations, are actually inverse processes. On the AP exam, you're being tested on your ability to move fluidly between these two operations: recognizing when to differentiate an accumulation function, when to evaluate a definite integral using antiderivatives, and how to interpret both in applied contexts.

These theorems unlock everything from computing areas under curves to solving motion problems to analyzing rates of change in real-world scenarios. You'll encounter them in accumulation function problems, net change applications, and average value calculations. Don't just memorize the formulas—understand why differentiation undoes integration and vice versa. That conceptual grasp will carry you through FRQs that ask you to interpret results, not just compute them.

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The Two Fundamental Theorems

These theorems establish the inverse relationship between differentiation and integration—the cornerstone of calculus.

First Fundamental Theorem of Calculus (Evaluation Theorem)

• Connects antiderivatives to definite integrals—if $f$ is continuous on $[a, b]$ and $F$ is any antiderivative of $f$, then $\int_a^b f(x)\,dx = F(b) - F(a)$
• Eliminates the need for Riemann sums in most calculations; instead of computing limits of sums, you simply evaluate the antiderivative at the bounds
• Requires continuity on the closed interval—this condition appears frequently in AP multiple choice questions asking when the theorem applies

Second Fundamental Theorem of Calculus (Derivative of Accumulation Functions)

• Defines accumulation functions as $F(x) = \int_a^x f(t)\,dt$, where the upper limit is the variable and $a$ is a constant
• States that $\frac{d}{dx}\int_a^x f(t)\,dt = f(x)$—differentiation and integration cancel when set up this way
• Extends via chain rule to $\frac{d}{dx}\int_a^{g(x)} f(t)\,dt = f(g(x)) \cdot g'(x)$, a high-frequency AP question type

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Evaluating Definite Integrals

The practical payoff of the First Fundamental Theorem: computing exact values without limits of sums.

Antiderivative Method for Definite Integrals

• Find any antiderivative $F(x)$ of the integrand $f(x)$—you don't need the $+C$ since it cancels in $F(b) - F(a)$
• Apply the evaluation formula $\int_a^b f(x)\,dx = F(b) - F(a)$, often written with bracket notation as $[F(x)]_a^b$
• Watch for discontinuities—if $f$ has a discontinuity in $[a, b]$, you may need to split the integral or the theorem doesn't directly apply

Definite Integrals as Signed Area

• Represents net area between the curve $y = f(x)$ and the $x$-axis over $[a, b]$—regions above the axis contribute positive area, regions below contribute negative
• Net vs. total distinction is critical: $\int_a^b f(x)\,dx$ gives net area, while $\int_a^b |f(x)|\,dx$ gives total area
• Geometric interpretation helps verify answers—sketch the region when possible to catch sign errors

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Accumulation Functions and the Second Theorem

When the upper limit of integration is a variable, you're building a function that measures accumulated change.

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

• Measures total accumulation of the quantity whose rate is $f(t)$, starting from the reference point $t = a$
• Always equals zero at the lower bound: $F(a) = \int_a^a f(t)\,dt = 0$, which often serves as an initial condition
• Inherits properties from $f$—if $f > 0$, then $F$ is increasing; if $f$ changes sign, $F$ has local extrema

Chain Rule Extension

• For composite upper limits, apply $\frac{d}{dx}\int_a^{g(x)} f(t)\,dt = f(g(x)) \cdot g'(x)$—this is the Second FTC combined with the chain rule
• For variable lower limits, use $\frac{d}{dx}\int_{h(x)}^{b} f(t)\,dt = -f(h(x)) \cdot h'(x)$—the negative sign comes from flipping the bounds
• Both limits variable requires splitting: $\frac{d}{dx}\int_{h(x)}^{g(x)} f(t)\,dt = f(g(x)) \cdot g'(x) - f(h(x)) \cdot h'(x)$

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Average Value and the Mean Value Theorem for Integrals

These results connect the definite integral to a single representative value of the function.

Mean Value Theorem for Integrals

• Guarantees existence of at least one $c$ in $(a, b)$ where $f(c) = \frac{1}{b-a}\int_a^b f(x)\,dx$, provided $f$ is continuous on $[a, b]$
• Geometric meaning: there's a rectangle with base $(b - a)$ and height $f(c)$ whose area equals the area under the curve
• Connects to average value—the value $f(c)$ is precisely the average value of $f$ over the interval

Average Value of a Function

• Formula: $f_{\text{avg}} = \frac{1}{b-a}\int_a^b f(x)\,dx$—memorize this; it appears frequently in applied contexts
• Interpretation: if $f$ represents a rate (like velocity or flow rate), the average value tells you the constant rate that would produce the same total accumulation
• Units matter—average value has the same units as $f(x)$, not the same units as the integral

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Applications: Net Change and Accumulation

The theorems translate directly into real-world contexts involving rates and total quantities.

Net Change Theorem

• States that $\int_a^b f'(x)\,dx = f(b) - f(a)$—the integral of a rate of change gives the net change in the original quantity
• Applies to motion: integrating velocity gives displacement, integrating acceleration gives change in velocity
• Applies to any rate: flow rate → total volume, marginal cost → total cost change, population growth rate → population change

Interpreting Integrals in Context

• Match units carefully: if $f(t)$ has units of gallons/minute and $t$ is in minutes, then $\int_a^b f(t)\,dt$ has units of gallons
• Signed results have meaning—a negative integral might indicate net outflow, net loss, or displacement in the negative direction
• Initial value problems combine the Net Change Theorem with given starting conditions: $f(b) = f(a) + \int_a^b f'(x)\,dx$

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Quick Reference Table

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Self-Check Questions

1. If $F(x) = \int_2^{x^3} \cos(t)\,dt$, what is $F'(x)$? Which theorem and rule did you use?

2. Compare and contrast: How does evaluating $\int_0^5 v(t)\,dt$ differ from evaluating $\int_0^5 |v(t)|\,dt$ when $v(t)$ changes sign? What physical quantities does each represent?

3. A function $f$ is continuous on $[1, 4]$ with $\int_1^4 f(x)\,dx = 12$. What is the average value of $f$ on this interval, and what does the Mean Value Theorem for Integrals guarantee?

4. Given that $g(x) = \int_x^{10} f(t)\,dt$, explain why $g'(x) = -f(x)$. How does this relate to the Second Fundamental Theorem?

5. An FRQ states: "The rate at which water flows into a tank is given by $r(t)$ gallons per minute. At $t = 0$, the tank contains 50 gallons." Write an expression for the amount of water in the tank at time $t = T$, and identify which theorem justifies your answer.