Fundamental Theorem of Calculus
Connects differentiation and integration.
Formula
Variables
Example
∫₀² 2x dx = [x²]₀² = 4 − 0 = 4.
How it works
This formula says that if F is any antiderivative of f, then the total accumulated change of f over an interval from a to b is found by subtracting endpoint values of F. It works because integration adds up infinitely many tiny slices, while differentiation measures how a function changes locally; the Fundamental Theorem of Calculus shows these two processes are inverse operations when they are matched correctly. In practice, it lets you evaluate definite integrals without approximating area piece by piece.
When to use it
Use this whenever you need the exact value of a definite integral and you can find an antiderivative of the integrand. It appears constantly in physics, economics, probability, and engineering whenever you need accumulated quantity from a rate function.
Worked example, step by step
Evaluate ∫₁³ (4x^3 - 2x) dx Step 1: Find an antiderivative F(x). F(x) = x^4 - x^2 Step 2: Apply the limits. F(3) - F(1) = (3^4 - 3^2) - (1^4 - 1^2) Step 3: Simplify each term. (81 - 9) - (1 - 1) = 72 - 0 Step 4: Final answer. ∫₁³ (4x^3 - 2x) dx = 72
Common mistakes to avoid
- Forgetting to subtract F(a) from F(b), not just evaluating at the upper limit
- Using the original function f(x) instead of an antiderivative F(x)
- Dropping constants or mishandling signs when evaluating the endpoints
Did You Know?
The fundamental theorem of calculus revealed that differentiation and integration are inverse operations — uniting the whole subject.
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