Math121 calculus 1 integration and integral
note
School: Collin County Community College,
texas
The definite integral
integration and integral note
Definition:The definite integral
Let f (x) be a function defined on a closed
interval (a,b). we say that a number (I) is the
definite integral of (f) over (a,b)and that (I) is
n
the limit of Riemann sums ∑ ¿1 f ¿ ¿) ∆ xk k
→
Notation and existence of the definite integral
The symbol for the number I in the definition
b
of the definite integral is ∫ f (x )dx which is read a
as the integral form a to b of f of x dee x . the
component parts in the integral symbol also
have name
b → upperlimit of integration
∫ f (x )dx
a → lower limit of integration
the function is the integral
f (x)→
When the definition is satisfied we say the
Riemann sums of on (a,b) converge the
b
definite I¿ ∫ ( fx ) dx and that f is integrable over
a
(a,b) we have many choices for partition P
, with norm going to zero any many choices of
point Ck for each partition The definite
integral exists when we always get the same
limit I, no matter what choices are made.
When the limit exist we write it as the definite
integral .
b
∑ ¿1 f ¿ ¿) ∆ ∆xk ¿ I =∫ f ( x ) dx
n
→
k a
When each partition has n equal sub interval
each of width ∆ x=¿ b-a) we also write
b
∑ ¿1 f ¿ ¿) ❑ ∆ xk ¿ I=∫ f ( x ) dx
n
→
k a
The value of the definite integral of a function
over any particular interval depends o the
function not on the letter we choose to
represent its indepedent variable. If we decide
to use t or u instead of x we simply write the
integral as
∫ f ( f ) dt∨¿ ¿ ∫ f ( u ) duintead of ∫ f ( x ) dx
b b b
a a a
No matter how we write the intergral it is still
the same number defined as a limit of
Riemann sums. since it does not matter what
letter we use. The variable of integration is
called dummy variable.
Theorem 1: A continuous function is
integrable. That is if function f is continuous
note
School: Collin County Community College,
texas
The definite integral
integration and integral note
Definition:The definite integral
Let f (x) be a function defined on a closed
interval (a,b). we say that a number (I) is the
definite integral of (f) over (a,b)and that (I) is
n
the limit of Riemann sums ∑ ¿1 f ¿ ¿) ∆ xk k
→
Notation and existence of the definite integral
The symbol for the number I in the definition
b
of the definite integral is ∫ f (x )dx which is read a
as the integral form a to b of f of x dee x . the
component parts in the integral symbol also
have name
b → upperlimit of integration
∫ f (x )dx
a → lower limit of integration
the function is the integral
f (x)→
When the definition is satisfied we say the
Riemann sums of on (a,b) converge the
b
definite I¿ ∫ ( fx ) dx and that f is integrable over
a
(a,b) we have many choices for partition P
, with norm going to zero any many choices of
point Ck for each partition The definite
integral exists when we always get the same
limit I, no matter what choices are made.
When the limit exist we write it as the definite
integral .
b
∑ ¿1 f ¿ ¿) ∆ ∆xk ¿ I =∫ f ( x ) dx
n
→
k a
When each partition has n equal sub interval
each of width ∆ x=¿ b-a) we also write
b
∑ ¿1 f ¿ ¿) ❑ ∆ xk ¿ I=∫ f ( x ) dx
n
→
k a
The value of the definite integral of a function
over any particular interval depends o the
function not on the letter we choose to
represent its indepedent variable. If we decide
to use t or u instead of x we simply write the
integral as
∫ f ( f ) dt∨¿ ¿ ∫ f ( u ) duintead of ∫ f ( x ) dx
b b b
a a a
No matter how we write the intergral it is still
the same number defined as a limit of
Riemann sums. since it does not matter what
letter we use. The variable of integration is
called dummy variable.
Theorem 1: A continuous function is
integrable. That is if function f is continuous
