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Fundamental Theorem
of Calculus

Master the connection between derivatives and integrals through the two parts of the FTC and learn to evaluate definite integrals with confidence.

AP Calculus AB

What is a Definite Integral?

A definite integral computes the net signed area between a function \( f(x) \) and the \( x \)-axis over a closed interval \( [a, b] \). Unlike an indefinite integral, the result is a number, not a family of functions.

x y a b Area f(x)

From Riemann Sums to the FTC

Before the FTC, the only way to evaluate \( \displaystyle\int_a^b f(x)\,dx \) was to compute a limit of Riemann sums: partition the interval, sum rectangle areas, and take the limit as the number of rectangles approaches infinity.

Before FTC
Evaluate \( \displaystyle\lim_{n\to\infty}\sum_{i=1}^{n}f(x_i^*)\Delta x \): tedious algebra for every integral.
After FTC
Find one antiderivative \( F(x) \) of \( f(x) \), then compute \( F(b) - F(a) \). Done.

Why the FTC Matters

The Fundamental Theorem of Calculus establishes that differentiation and integration are inverse operations. This single idea unifies the two main branches of calculus and appears in nearly every application:

Physics
Velocity \( \to \) displacement via definite integrals of rate functions.
Economics
Marginal cost \( \to \) total cost by integrating over a production interval.
Engineering
Compute work, fluid force, and accumulated quantities using definite integrals.

FTC Part 1: The Accumulation Function

In words: the derivative of the integral (with a variable upper limit) gives back the original function. This proves that every continuous function has an antiderivative.

FTC Part 2: The Evaluation Theorem

The notation \( F(x)\Big|_a^b \) or \( \bigl[F(x)\bigr]_a^b \) means "evaluate \( F \) at the upper limit and subtract its value at the lower limit."

x y a b F(a) F(b)
\( F(b)-F(a) \)
 F(x)

Part 1 vs Part 2: Key Distinction

FTC Part 1 FTC Part 2
Statement \( \frac{d}{dx}\!\int_a^x f(t)\,dt = f(x) \) \( \int_a^b f(x)\,dx = F(b) - F(a) \)
Direction Differentiation undoes integration Antiderivative evaluates an integral
Output A function \( f(x) \) A number
Upper limit Variable \( x \) Constant \( b \)
Use case Find derivatives of integral expressions Evaluate definite integrals

How to Apply FTC Part 2

1
Verify continuity. Confirm \( f(x) \) is continuous on \( [a, b] \). If not, the FTC cannot be applied directly.
2
Find an antiderivative. Determine \( F(x) \) such that \( F'(x) = f(x) \). You only need one antiderivative: no \( +C \) needed.
3
Evaluate at the bounds. Compute \( F(b) \) and \( F(a) \).
4
Subtract. The answer is \( F(b) - F(a) \). Always subtract the lower limit value from the upper limit value.

Example 1: Evaluate \( \displaystyle\int_1^4 2x\,dx \)

1
Check continuity: \( f(x) = 2x \) is a polynomial: continuous everywhere. ✓
2
Antiderivative: \( F(x) = x^2 \) since \( \frac{d}{dx}[x^2] = 2x \).
3
Evaluate: \( F(4) = 16 \) and \( F(1) = 1 \).
4
Subtract: \( 16 - 1 = \) 15.

Visualizing \( \displaystyle\int_1^4 2x\,dx = \) 15

x y 1 2 3 4 2 4 6 8 Area = 15 f(x) = 2x

Example 2: Evaluate \( \displaystyle\int_0^{\pi} \sin x\,dx \)

1
Continuity: \( \sin x \) is continuous on \( [0, \pi] \). ✓
2
Antiderivative: \( F(x) = -\cos x \) since \( \frac{d}{dx}[-\cos x] = \sin x \).
3
Evaluate: \( F(\pi) = -\cos(\pi) = -(-1) = 1 \) and \( F(0) = -\cos(0) = -1 \).
4
Subtract: \( 1 - (-1) = \) 2.

Visualizing \( \displaystyle\int_0^{\pi} \sin x\,dx = \) 2

x y 0
\(\frac{\pi}{2}\)
\(\pi\)
\(\frac{3\pi}{2}\)
 1 Area = 2
\( \sin x \)

The shaded region under one full arch of \( \sin x \) from \( 0 \) to \( \pi \) has an area of exactly 2 square units.

Example 3: Evaluate \( \displaystyle\int_1^{e^2} \frac{3}{x}\,dx \)

1
Continuity: \( \frac{3}{x} \) is continuous on \( [1, e^2] \) since \( x > 0 \) throughout. ✓
2
Antiderivative: \( F(x) = 3\ln|x| = 3\ln x \) (since \( x > 0 \)).
3
Evaluate: \( F(e^2) = 3\ln(e^2) = 3 \cdot 2 = 6 \) and \( F(1) = 3\ln(1) = 0 \).
4
Subtract: \( 6 - 0 = \) 6.

Applying FTC Part 1 with Chain Rule

When the upper limit is a function \( u(x) \) instead of just \( x \), combine FTC Part 1 with the Chain Rule:

Example: Find \( \displaystyle\frac{d}{dx}\int_2^{x^3}\!\sqrt{t+1}\,dt \).

1
Identify \( f(t) = \sqrt{t+1} \) and \( u(x) = x^3 \), so \( u'(x) = 3x^2 \).
2
Apply the formula: \( f(u(x)) \cdot u'(x) = \sqrt{x^3+1}\cdot 3x^2 \).

Common Mistakes to Avoid

✕ Forgetting the negative sign
The antiderivative of \( \sin x \) is \( -\cos x \), not \( \cos x \). Always verify by differentiating your antiderivative.
✕ Subtracting in wrong order
It is \( F(b) - F(a) \), never \( F(a) - F(b) \). Upper limit first, lower limit subtracted.
✕ Adding \( +C \) to definite integrals
The constant cancels: \( (F(b)+C)-(F(a)+C) = F(b)-F(a) \). Never include \( +C \) in a definite integral.

Summary of Key Formulas

Scenario Formula Result Type
Definite integral evaluation \( \int_a^b f(x)\,dx = F(b)-F(a) \) Number
Accumulation function derivative \( \frac{d}{dx}\int_a^x f(t)\,dt = f(x) \) Function
Chain rule variant \( \frac{d}{dx}\int_a^{u(x)} f(t)\,dt = f(u)\cdot u' \) Function
Reversed limits \( \int_b^a f(x)\,dx = -\int_a^b f(x)\,dx \) Number
Splitting intervals \( \int_a^b f\,dx = \int_a^c f\,dx + \int_c^b f\,dx \) Number

Question 1 of 5

Evaluate \( \displaystyle\int_0^3 (4x - 1)\,dx \).

Question 2 of 5

What is \( \displaystyle\int_0^{\pi/2}\cos x\,dx \)?

Question 3 of 5

Using FTC Part 1, find \( \displaystyle\frac{d}{dx}\int_1^x (t^2 + 3t)\,dt \).

Question 4 of 5

Evaluate \( \displaystyle\int_1^e \frac{5}{x}\,dx \).

Question 5 of 5

Find \( \displaystyle\frac{d}{dx}\int_0^{x^2}\!\sin(t)\,dt \).