Definite Integrals and Their Properties
The definite integral of f from a to b equals F(b) - F(a) for any antiderivative F, and satisfies additivity, reversal and symmetry properties.
What is Definite Integrals and Their Properties?
The definite integral of f from a to b equals F(b) - F(a) for any antiderivative F, and satisfies additivity, reversal and symmetry properties.
Key formula / rule: Reversing the limits changes the sign of the integral.
Key points
- Apply symmetry properties to evaluate definite integrals.
- Compute definite integrals with correctly transformed limits.
Common exam trap
Ignoring a discontinuity of the integrand inside the interval.
Definitions
- Term
Definite Integrals and Their Properties
- Meaning
The definite integral of f from a to b equals F(b) - F(a) for any antiderivative F, and satisfies additivity, reversal and symmetry properties.
- Term
Definite Integrals and Their Properties — explanation
- Meaning
The fundamental theorem links area to antiderivatives, and the symmetry properties often evaluate an integral without any antiderivative at all. King's rule is the most examined of these.
Learning objectives
Apply symmetry properties to evaluate definite integrals.
Compute definite integrals with correctly transformed limits.
Formulae
- Key point
Reversing the limits changes the sign of the integral.
- Key point
The integral of an odd function over a symmetric interval is zero.
- Key point
King's rule replaces x by a + b - x without changing the value.
Prerequisites
MAT-U2-IC-T1-S1-C2
Common mistakes
Ignoring a discontinuity of the integrand inside the interval.
Keeping the original limits after a substitution.
Keywords
Definite
Integrals
Their
Properties
Practice preview
What is the value of ∫ from 0 to 3 of 5 dx?…
easy
Evaluate ∫ from pi/6 to pi/3 of (1 / (1 + sqrt(tan x))) dx.…
medium
Evaluate ∫ from 0 to 2pi of |cos x| dx.…
hard