Math & Statistics · 积分（Integral） · Updated 2026-10-25
What Is Integral?
Definition
The definite integral ∫[a,b] f(x)dx = lim Σ f(xi)Δx as partitions refine — the signed area between curve and axis. The Newton–Leibniz formula links it to antiderivatives: ∫[a,b] f(x)dx = F(b)−F(a) with F'=f, turning "find the area" into "find F and subtract." Velocity integrated over time is displacement (area under v-t), power over time is energy, marginal revenue integrated is total revenue: every cumulative effect is an integral.
Formula
∫[a,b] f(x)dx = F(b) − F(a), where F'(x) = f(x)
Worked Example
With v(t)=3t² m/s, displacement from 0 to 2 s is ∫0..2 3t²dt = 8 m; a constant 2 kW load over 3 hours consumes 6 kWh.
Common Misconception
Reading ∫ as ordinary multiplication — dx is the partition variable, not "multiply by dx": ∫0..2 x dx = 2, not 2×2 = 4. Another trap: below the axis the integral is negative; physical area needs absolute value or piecewise handling.
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