Area as Definite Integral
Area as Definite Integral:

Objective:
 The definite integral of a function $f(x)$ from $a$ to $b$ represents the area enclosed by the curve $y=f(x)$, the xaxis, and the lines $x=a$ and $x=b$.

Formulation:
 The area $A$ under $f(x)$ from $x=a$ to $x=b$ is denoted by: $A={\int}_{a}^{b}f(x)\text{\hspace{0.17em}}dx$
 Here, $f(x)$ should be continuous and nonnegative over $[a,b]$.

Geometric Interpretation:
 For $f(x)\ge 0$ over $[a,b]$, the definite integral represents the accumulation of infinitely small vertical strips of width $dx$ and height $f(x)$.
 The summation of these strips over $[a,b]$ gives the total area.

Area between Curve and xaxis:
 If $f(x)\ge 0$ for $x$ in $[a,b]$, the definite integral gives the area between the curve and the xaxis over that interval.

Area Enclosed by Curves:
 When finding the area between two curves $f(x)$ and $g(x)$ over $[a,b]$, the area is given by: $A={\int}_{a}^{b}\mid f(x)g(x)\mid \text{\hspace{0.17em}}dx$ where $\mid f(x)g(x)\mid $represents the vertical distance between the curves.

Negative Area:
 If $f(x)\le 0$ over $[a,b]$, the integral ${\int}_{a}^{b}f(x)\text{\hspace{0.17em}}dx$ gives the negative of the area between the curve and the xaxis.

Applications:
 Crucial in various fields like physics, economics, engineering for calculating quantities such as total displacement, work done, and accumulated values over time.

Visualization and Interpretation:
 Graphical representations and visualizations aid in understanding the relationship between the definite integral and area under curves.
 Riemann sums provide a way to approximate area and understand the process behind definite integrals.