Properties of Definite Integrals
RELATED CALCULATOR: Definite and Improper Integral Calculator
Now let's see what properties integral has.
a
Property 1. It follows from the definition of integral that ∫a f (x)dx = 0.
b a
Property 2. Inverting bounds of integration: ∫a f (x)dx = −∫b f (x)dx.
a a
Property 3. If f is an even function then ∫−a f (x)dx = 2∫0 f (x)dx.
In terms of areas, this means the following: if f is symmetric about the x-axis, then the area from
−a to 0 equals the area from 0 to a.
a
Property 4. If f is an odd function then ∫−a f (x)dx = 0.
In terms of areas, this means the following: if f is symmteric about the origin, then the area from
−a to 0 equals minus area from 0 to a. This means that net area is 0.
b c b
Integral Adjacency: ∫a f (x)dx = ∫a + ∫c f (x)dx for any displacement of points a, b and c.
This study source was downloaded by 100000898062787 from CourseHero.com on 09-29-2025 04:46:21 GMT -05:00
https://www.coursehero.com/file/250856145/definite-integral-propertiespdf/
RELATED CALCULATOR: Definite and Improper Integral Calculator
Now let's see what properties integral has.
a
Property 1. It follows from the definition of integral that ∫a f (x)dx = 0.
b a
Property 2. Inverting bounds of integration: ∫a f (x)dx = −∫b f (x)dx.
a a
Property 3. If f is an even function then ∫−a f (x)dx = 2∫0 f (x)dx.
In terms of areas, this means the following: if f is symmetric about the x-axis, then the area from
−a to 0 equals the area from 0 to a.
a
Property 4. If f is an odd function then ∫−a f (x)dx = 0.
In terms of areas, this means the following: if f is symmteric about the origin, then the area from
−a to 0 equals minus area from 0 to a. This means that net area is 0.
b c b
Integral Adjacency: ∫a f (x)dx = ∫a + ∫c f (x)dx for any displacement of points a, b and c.
This study source was downloaded by 100000898062787 from CourseHero.com on 09-29-2025 04:46:21 GMT -05:00
https://www.coursehero.com/file/250856145/definite-integral-propertiespdf/
