Definite Integral

analysis

Definition

Definite Integral

Let $I=[a,b]$ be an interval and $f:I\to \mathbb{R}$ a function. If every sequence $(S_n{)}_{n\in \mathbb{N}}$ of Riemann sums, whose corresponding sequence of partitions satisfies ${\lim }_{n\to \infty }\text{mesh}(Z_n)=0$, converges to the same limit, then this limit is called the definite integral of $f$ on $[a,b]$ and is denoted by:

$${\int }_a^{b}f(x)dx$$

Geometrically, for a non-negative function, the definite integral represents the signed area under the curve between the limits of integration $a$ and $b$.

Boundary Properties

If the limits of integration are equal or reversed, one defines:

$${\int }_a^{a}f(x)dx=0\text{and}{\int }_a^{b}f(x)dx=-{\int }_b^{a}f(x)dx$$

Integration Techniques

Substitution

Let $f$ be continuous on $[a,b]$, and $g:[c,d]\to [a,b]$ a differentiable function with $g(c)=a$ and $g(d)=b$. Then:

$${\int }_a^{b}f(u)du={\int }_c^{d}f(g(x))\cdot g^{\prime }(x)dx$$

Integration by Parts

If $u$ and $v$ are continuously differentiable on $[a,b]$, then:

$${\int }_a^{b}u(x)v^{\prime }(x)dx=[u(x)v(x)]\_a^b-{\int }_a^b{u}^{\prime }(x)v(x)dx$$