Calculus
Class 12

Definite Integral

abf(x)dx=F(b)F(a)\int_a^b f(x)dx = F(b) - F(a)
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∫[a to b]f(x)dx = F(b) - F(a)

Adjust Variables

Lower bound a
a =
05
Upper bound b
b =
110

The Fundamental Theorem of Calculus. Definite integral equals antiderivative evaluated at bounds.

Real-World Applications

Physics — Total distance traveled: ∫v dt from t₁ to t₂.

Engineering — Area under load-deflection curves.

Economics — Consumer/producer surplus calculations.

Statistics — Probability between two values: P(a<X<b) = ∫f(x)dx.

Medicine — Total drug exposure (AUC - Area Under Curve).

Hydrology — Total water flow over a time period.

Environmental — Total pollution released.

Manufacturing — Total production over a shift.

Finance — Present value of cash flow stream.

Physics — Work done by variable force.

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