Limits of integration explained

In calculus and mathematical analysis the limits of integration (or bounds of integration) of the integral$$\backslash int\_a^b\; f(x)\; \backslash ,\; dx$$

f defined on a closed and bounded interval are the real numbers a and b , in which a is called the lower limit and b the upper limit. The region that is bounded can be seen as the area inside a and b .

For example, the function

f(x) = x ^{3} is defined on the interval [ 2 , 4 ] $$\backslash int\_2^4\; x^3\; \backslash ,\; dx$$with the limits of integration being 2 and 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 and b are solved for f(u) . In general,$$\backslash int\_a^b\; f(g(x))g\text{'}(x)\; \backslash \; dx\; =\; \backslash int\_^\; f(u)\; \backslash \; du$$where u = g(x) and du = g'(x) dx . Thus, a and b will be solved in terms of u ; the lower bound is g(a) and the upper bound is g(b) .

For example,$$\backslash int\_0^2\; 2x\backslash cos(x^2)dx\; =\; \backslash int\_0^4\backslash cos(u)\; \backslash ,\; du$$

where

u = x ^{2} and du =2 xdx . Thus, f( 0 ) =0 ^{2=0} and f( 2 ) =2 ^{2=4} . Hence, the new limits of integration are 0 and 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$$\backslash lim\_\; \backslash int\_z^b\; f(x)\; \backslash ,\; dx$$and$$\backslash lim\_\; \backslash int\_a^z\; f(x)\; \backslash ,\; dx$$again being a and b. For an improper integral$$\backslash int\_a^\backslash infty\; f(x)\; \backslash ,\; dx$$or$$\backslash int\_^b\; f(x)\; \backslash ,\; dx$$the limits of integration are a and ∞, or -∞ and b, respectively.^[3]

Definite Integrals

If

c\in(a,b) , then^[4] $$\backslash int\_a^b\; f(x)\backslash \; dx\; =\; \backslash int\_a^c\; f(x)\backslash \; dx\; \backslash \; +\; \backslash int\_c^b\; f(x)\backslash \; dx.$$

See also

• Integral
• Riemann integration
• Definite integral

Notes and References

1. Web site: 31.5 Setting up Correct Limits of Integration. math.mit.edu. 2019-12-02.
2. Web site: -substitution. Khan Academy. en. 2019-12-02.
3. Web site: Calculus II - Improper Integrals. tutorial.math.lamar.edu . 2019-12-02.
4. Web site: Definite Integral . Weisstein. Eric W.. mathworld.wolfram.com. en. 2019-12-02.