# Area as Definite Integral

### Area as Definite Integral:

1. Objective:

• The definite integral of a function $f\left(x\right)$ from $a$ to $b$ represents the area enclosed by the curve $y=f\left(x\right)$, the x-axis, and the lines $x=a$ and $x=b$.
2. Formulation:

• The area $A$ under $f\left(x\right)$ from $x=a$ to $x=b$ is denoted by: $A={\int }_{a}^{b}f\left(x\right)\text{\hspace{0.17em}}dx$
• Here, $f\left(x\right)$ should be continuous and non-negative over $\left[a,b\right]$.
3. Geometric Interpretation:

• For $f\left(x\right)\ge 0$ over $\left[a,b\right]$, the definite integral represents the accumulation of infinitely small vertical strips of width $dx$ and height $f\left(x\right)$.
• The summation of these strips over $\left[a,b\right]$ gives the total area.
4. Area between Curve and x-axis:

• If $f\left(x\right)\ge 0$ for $x$ in $\left[a,b\right]$, the definite integral gives the area between the curve and the x-axis over that interval.
5. Area Enclosed by Curves:

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

• If $f\left(x\right)\le 0$ over $\left[a,b\right]$, the integral ${\int }_{a}^{b}f\left(x\right)\text{\hspace{0.17em}}dx$ gives the negative of the area between the curve and the x-axis.
7. Applications:

• Crucial in various fields like physics, economics, engineering for calculating quantities such as total displacement, work done, and accumulated values over time.
8. 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.