Upper and lower limits applied in definite integration
In calculus and mathematical analysis the limits of integration (or bounds of integration ) of the integral
∫
a
b
f
(
x
)
d
x
{\displaystyle \int _{a}^{b}f(x)\,dx}
of a Riemann integrable function
f
{\displaystyle f}
defined on a closed and bounded interval are the real numbers
a
{\displaystyle a}
and
b
{\displaystyle b}
, in which
a
{\displaystyle a}
is called the lower limit and
b
{\displaystyle b}
the upper limit . The region that is bounded can be seen as the area inside
a
{\displaystyle a}
and
b
{\displaystyle b}
.
For example, the function
f
(
x
)
=
x
3
{\displaystyle f(x)=x^{3}}
is defined on the interval
[
2
,
4
]
{\displaystyle [2,4]}
∫
2
4
x
3
d
x
{\displaystyle \int _{2}^{4}x^{3}\,dx}
with the limits of integration being
2
{\displaystyle 2}
and
4
{\displaystyle 4}
.
[1]
Integration by Substitution (U-Substitution)
In Integration by substitution , the limits of integration will change due to the new function being integrated. With the function that is being derived,
a
{\displaystyle a}
and
b
{\displaystyle b}
are solved for
f
(
u
)
{\displaystyle f(u)}
. In general,
∫
a
b
f
(
g
(
x
)
)
g
′
(
x
)
d
x
=
∫
g
(
a
)
g
(
b
)
f
(
u
)
d
u
{\displaystyle \int _{a}^{b}f(g(x))g'(x)\ dx=\int _{g(a)}^{g(b)}f(u)\ du}
where
u
=
g
(
x
)
{\displaystyle u=g(x)}
and
d
u
=
g
′
(
x
)
d
x
{\displaystyle du=g'(x)\ dx}
. Thus,
a
{\displaystyle a}
and
b
{\displaystyle b}
will be solved in terms of
u
{\displaystyle u}
; the lower bound is
g
(
a
)
{\displaystyle g(a)}
and the upper bound is
g
(
b
)
{\displaystyle g(b)}
.
For example,
∫
0
2
2
x
cos
(
x
2
)
d
x
=
∫
0
4
cos
(
u
)
d
u
{\displaystyle \int _{0}^{2}2x\cos(x^{2})dx=\int _{0}^{4}\cos(u)\,du}
where
u
=
x
2
{\displaystyle u=x^{2}}
and
d
u
=
2
x
d
x
{\displaystyle du=2xdx}
. Thus,
f
(
0
)
=
0
2
=
0
{\displaystyle f(0)=0^{2}=0}
and
f
(
2
)
=
2
2
=
4
{\displaystyle f(2)=2^{2}=4}
. Hence, the new limits of integration are
0
{\displaystyle 0}
and
4
{\displaystyle 4}
.[2]
The same applies for other substitutions.
Improper integrals
Limits of integration can also be defined for improper integrals , with the limits of integration of both
lim
z
→
a
+
∫
z
b
f
(
x
)
d
x
{\displaystyle \lim _{z\to a^{+}}\int _{z}^{b}f(x)\,dx}
and
lim
z
→
b
−
∫
a
z
f
(
x
)
d
x
{\displaystyle \lim _{z\to b^{-}}\int _{a}^{z}f(x)\,dx}
again being
a and
b . For an
improper integral
∫
a
∞
f
(
x
)
d
x
{\displaystyle \int _{a}^{\infty }f(x)\,dx}
or
∫
−
∞
b
f
(
x
)
d
x
{\displaystyle \int _{-\infty }^{b}f(x)\,dx}
the limits of integration are
a and ∞, or −∞ and
b , respectively.
[3]
Definite Integrals
If
c
∈
(
a
,
b
)
{\displaystyle c\in (a,b)}
, then[4]
∫
a
b
f
(
x
)
d
x
=
∫
a
c
f
(
x
)
d
x
+
∫
c
b
f
(
x
)
d
x
.
{\displaystyle \int _{a}^{b}f(x)\ dx=\int _{a}^{c}f(x)\ dx\ +\int _{c}^{b}f(x)\ dx.}
See also
References
^ "31.5 Setting up Correct Limits of Integration". math.mit.edu . Retrieved 2019-12-02 .
^ "𝘶-substitution". Khan Academy . Retrieved 2019-12-02 .
^ "Calculus II - Improper Integrals". tutorial.math.lamar.edu . Retrieved 2019-12-02 .
^ Weisstein, Eric W. "Definite Integral". mathworld.wolfram.com . Retrieved 2019-12-02 .