Math
Fundamental theorem.
Definition
The Fundamental Theorem of Calculus connects differentiation and integration. Part 1 states that if F(x) = ∫ from a to x of f(t) dt, then F′(x) = f(x). Part 2 states that ∫ from a to b of f(x) dx = F(b) − F(a), where F is any antiderivative of f.
Examples
Real-world.
- 1 To evaluate ∫ from 1 to 3 of 2x dx, find the antiderivative x² and compute 3² − 1² = 8
- 2 If F(x) = ∫ from 0 to x of cos(t) dt, then F′(x) = cos(x) by Part 1
Key Fact
∫ from a to b of f(x) dx = F(b) − F(a), where F′(x) = f(x)