Geometrical Interpretation of Definite Integral
Geometrical Interpretation of Definite Integrals:

Area Under a Curve:
 The definite integral represents the accumulated net area between a function and the xaxis over a specified interval $[a,b]$.
 The integral ${\int}_{a}^{b}f(x)\text{\hspace{0.17em}}dx$ computes the net area of the region bounded by $f(x)$, the xaxis, and the vertical lines $x=a$ and $x=b$.

Partitioning the Interval:
 To estimate area, partition the interval $[a,b]$ into $n$ subintervals.
 Width of each subinterval: $\mathrm{\Delta}x=\frac{ba}{n}$.

Riemann Sum:
 Approximates the area using rectangles whose heights are determined by the function values at specific points within each subinterval.
 The sum of the areas of these rectangles approaches the area under the curve as $n$ approaches infinity.

Definite Integral as Limit of Riemann Sums:
 As $n$ becomes infinitely large and $\mathrm{\Delta}x$ tends to zero, the Riemann sum approaches the definite integral: ${\int}_{a}^{b}f(x)\text{\hspace{0.17em}}dx={\mathrm{lim}}_{n\to \mathrm{\infty}}{\sum}_{i=1}^{n}f({x}_{i})\mathrm{\Delta}x$.

Positive and Negative Areas:
 Positive values of $f(x)$ contribute to positive areas above the xaxis.
 Negative values of $f(x)$ contribute to negative areas below the xaxis.

Interpretation with Functions:
a. Above and Below the xaxis: For $f(x)\ge 0$ over $[a,b]$, the definite integral represents the area above the xaxis.
b. Negative Function Values: When $f(x)\le 0$ over $[a,b]$, the integral represents the negative of the area below the xaxis.
Importance of Geometric Interpretation:

Visualization of Integrals: Provides a visual understanding of the concept of definite integrals as areas under curves.

Applications in Geometry: Facilitates computations of areas of irregular shapes and volumes of solids by using integrals.

Understanding Function Behavior: Helps in understanding the behavior of functions over specific intervals through their graphical representations.