Upper and lower limits applied in definite integration
In calculus and mathematical analysis the limits of integration (or bounds of integration) of the integral
![{\displaystyle \int _{a}^{b}f(x)\,dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac02adeed584466d53dee65f3228ad66939eb58b)
of a Riemann integrable function
defined on a closed and bounded interval are the real numbers
and
, in which
is called the lower limit and
the upper limit. The region that is bounded can be seen as the area inside
and
.
For example, the function
is defined on the interval
![{\displaystyle \int _{2}^{4}x^{3}\,dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/879dd2af234a6d319350b0ffe3859eaa2249200d)
with the limits of integration being
![{\displaystyle 2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f)
and
![{\displaystyle 4}](https://wikimedia.org/api/rest_v1/media/math/render/svg/295b4bf1de7cd3500e740e0f4f0635db22d87b42)
.
[1] Integration by Substitution (U-Substitution)[edit]
In Integration by substitution, the limits of integration will change due to the new function being integrated. With the function that is being derived,
and
are solved for
. In general,
![{\displaystyle \int _{a}^{b}f(g(x))g'(x)\ dx=\int _{g(a)}^{g(b)}f(u)\ du}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6de765d8ce8f05f6c9cf70d0201efd12948c9c95)
where
![{\displaystyle u=g(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54c52880c0cf28150ff774cb45cddcd2a027674f)
and
![{\displaystyle du=g'(x)\ dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4e9a6d80b0cc773db2f5d4bcad12d08acd18ed9)
. Thus,
![{\displaystyle a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc)
and
![{\displaystyle b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3)
will be solved in terms of
![{\displaystyle u}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8)
; the lower bound is
![{\displaystyle g(a)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7205153da0f7522baf23e2a109ba6630e8104c7)
and the upper bound is
![{\displaystyle g(b)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e350ee25ae84e5f7f959ceed95c0d9b0ff8a05fa)
.
For example,
![{\displaystyle \int _{0}^{2}2x\cos(x^{2})dx=\int _{0}^{4}\cos(u)\,du}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac19e8cb9e2221281d2f492b7a5bb1b0ca62406)
where
and
. Thus,
and
. Hence, the new limits of integration are
and
.[2]
The same applies for other substitutions.
Improper integrals[edit]
Limits of integration can also be defined for improper integrals, with the limits of integration of both
![{\displaystyle \lim _{z\to a^{+}}\int _{z}^{b}f(x)\,dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7cfbf39655db6426e9fdba667d815678bb32619d)
and
![{\displaystyle \lim _{z\to b^{-}}\int _{a}^{z}f(x)\,dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49f8b3a4a1eb18c99344fba7b4e2097766d3914a)
again being
a and
b. For an
improper integral ![{\displaystyle \int _{a}^{\infty }f(x)\,dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10cbfbdc4ec0d2bb5f6bf26cc8d1b9292485bcfc)
or
![{\displaystyle \int _{-\infty }^{b}f(x)\,dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/82a9aa92eebfdb5417a4895dcf6d5a62510fa0ff)
the limits of integration are
a and ∞, or −∞ and
b, respectively.
[3] Definite Integrals[edit]
If
, then[4]
![{\displaystyle \int _{a}^{b}f(x)\ dx=\int _{a}^{c}f(x)\ dx\ +\int _{c}^{b}f(x)\ dx.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/abe9d1b88c5386c0b27e6e6b7dca4c550e9305b4)
See also[edit]
References[edit]