integral calculus

A function is said to be integrable if its integral over a specified domain exists and is finite. There are different types of integrability depending on the type of integral being considered.

Riemann Integrability §

A function $f:[a,b]\to \mathbb{R}$ is Riemann integrable if the Riemann sum converges to the same value regardless of how the partition of the interval is chosen, provided the partition gets finer and finer. Formally, $f$ is Riemann integrable if:

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

exists, which means:

$$\underset{||P||\to 0}{\lim }\sum _{i=1}^{n}f(x_i^{\ast })\Delta x_i$$

is the same for all partitions $P$ where $||P||\to 0$ (the norm of the partition, which is the length of the largest subinterval, goes to zero).

Key Points for Riemann Integrability: §

1. Boundedness: $f$ must be bounded on ([a, b]).
2. Set of Discontinuities: The set of points where $f$ is discontinuous must have measure zero (the Lebesgue measure of the set of discontinuities is zero).

Lebesgue Integrability §

A function $f:\mathbb{R}\to \mathbb{R}$ is Lebesgue integrable if the Lebesgue integral of the absolute value of $f$ is finite. Formally, $f$ is Lebesgue integrable if:

$${\int }_{-\infty }^{\infty }|f(x)|d\mu <\infty ,$$

where $\mu$ is the Lebesgue measure.

Key Points for Lebesgue Integrability: §

1. Absolute Integrability: The integral of the absolute value of the function over the entire domain must be finite.
2. Measurability: The function $f$ must be measurable, which means it should be compatible with the measure space (typically, the Lebesgue measure on $\mathbb{R}$).

Area between curves §

$$A={\int }_a^b\left(\genfrac{}{}{0ex}{}{\text{ upper }}{\text{ function }}\right)-\left(\genfrac{}{}{0ex}{}{\text{ lower }}{\text{ function }}\right)dx,a\le x\le b$$

$$A={\int }_c^d\left(\genfrac{}{}{0ex}{}{\text{ right }}{\text{ function }}\right)-\left(\genfrac{}{}{0ex}{}{\text{ left }}{\text{ function }}\right)dy,c\le y\le d$$