Working Rules for Finding Area Using Definite Integrals
Working Rules for Finding Area Using Definite Integrals:

Positive Function:
 Rule 1: For a nonnegative function $f(x)\ge 0$ over an interval $[a,b]$, the area under the curve $y=f(x)$ from $x=a$ to $x=b$ is given by the definite integral: $\text{Area}={\int}_{a}^{b}f(x)\text{\hspace{0.17em}}dx$
 This rule applies when the function remains above the xaxis for the entire interval, representing the area between the curve and the xaxis.

Multiple Regions and Piecewise Functions:
 Rule 2: If the function $f(x)$ changes sign or has multiple regions over $[a,b]$, find the areas of each region separately and sum their absolute values to get the total area. $\text{TotalArea}=\mid {\int}_{a}^{c}f(x)\text{\hspace{0.17em}}dx\mid +\mid {\int}_{c}^{b}f(x)\text{\hspace{0.17em}}dx\mid $

 Here, $c$ is the point where $f(x)$ changes sign or the regions switch.

Area between Curves:
 Rule 3: When finding the area between two curves $f(x)$ and $g(x)$ over $[a,b]$, compute the definite integral of their absolute difference: $\text{Area}={\int}_{a}^{b}\mid f(x)g(x)\mid \text{\hspace{0.17em}}dx$
 This rule applies to determine the area enclosed by both curves within the given interval.

Negative Function:
 Rule 4: If $f(x)\le 0$ over $[a,b]$, the definite integral ${\int}_{a}^{b}f(x)\text{\hspace{0.17em}}dx$ gives the negative of the area between the curve $y=f(x)$ and the xaxis.
 The magnitude of this integral represents the area, but the negative sign indicates that the area lies below the xaxis.
 Rule 4: If $f(x)\le 0$ over $[a,b]$, the definite integral ${\int}_{a}^{b}f(x)\text{\hspace{0.17em}}dx$ gives the negative of the area between the curve $y=f(x)$ and the xaxis.

Visualization and Graphical Analysis:
 Rule 5: Graphical representations help visualize the regions and understand the behavior of functions with respect to the xaxis for accurate area calculation using definite integrals.
 Riemann sums or numerical approximations aid in understanding and approximating areas under curves.
Example:
Consider the function $f(x)={x}^{2}2x$ over the interval $[0,2]$. Let's find the area between the curve $y=f(x)$ and the xaxis within this interval.

Identify the Region:
 The function $f(x)={x}^{2}2x$ intersects the xaxis at $x=0$ and $x=2$.
 To find the area between the curve and the xaxis, we'll compute the integral ${\int}_{0}^{2}\mid f(x)\mid \text{\hspace{0.17em}}dx$.

Integral Calculation: $\text{Area}={\int}_{0}^{2}\mid {x}^{2}2x\mid \text{\hspace{0.17em}}dx$
 Factorizing: $\mid {x}^{2}2x\mid =\mathrm{\mid}x(x2)\mathrm{\mid}$
 Over $[0,2]$, $x(x2)$ is positive, so $\mid {x}^{2}2x\mid =x(x2)$

Integral Setup and Solution: $\text{Area}={\int}_{0}^{2}x(x2)\text{\hspace{0.17em}}dx$
 Compute the integral: $\text{Area}={[\frac{{x}^{3}}{3}{x}^{2}]}_{0}^{2}=[\frac{{2}^{3}}{3}{2}^{2}][\frac{{0}^{3}}{3}{0}^{2}]$$\text{Area}=[\frac{8}{3}4][00]=\frac{8}{3}4=\frac{812}{3}=\frac{4}{3}$

Result:
 The area between the curve $y={x}^{2}2x$ and the xaxis over $[0,2]$ is $\frac{4}{3}$ square units.
 The negative sign indicates that this area lies below the xaxis due to the nature of the function.