Integration Explorer
Visualize definite integrals as shaded area, explore the Fundamental Theorem of Calculus, and compute area between two curves.
Integration
Values
| Function | -- |
| Bounds | -- |
| Numerical ∫ | -- |
| Exact value | -- |
| Antiderivative | -- |
| FTC | -- |
The Math Behind It
Definite Integrals
- Defined as the limit of Riemann sums: ∫ab f(x)dx = limn→∞ ∑ f(xi)Δx
- Notation: ∫ab f(x)dx where a is the lower limit and b is the upper limit
- Geometric meaning: the signed area between f(x) and the x-axis from a to b
- Area above the x-axis is positive; area below is negative
- Properties: linearity, additivity over intervals, comparison
Fundamental Theorem of Calculus
- FTC Part 1: d/dx ∫ax f(t)dt = f(x) — differentiation undoes integration
- FTC Part 2: ∫ab f(x)dx = F(b) − F(a) where F is any antiderivative of f
- Connects the two central operations of calculus: differentiation and integration
- Area between curves: ∫ab |f(x) − g(x)|dx
- When f(x) ≥ g(x) on [a, b], the absolute value can be dropped
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