# Geometrical Interpretation of Definite Integral

### Geometrical Interpretation of Definite Integrals:

1. Area Under a Curve:

• The definite integral represents the accumulated net area between a function and the x-axis over a specified interval $\left[a,b\right]$.
• The integral ${\int }_{a}^{b}f\left(x\right)\text{\hspace{0.17em}}dx$ computes the net area of the region bounded by $f\left(x\right)$, the x-axis, and the vertical lines $x=a$ and $x=b$.
2. Partitioning the Interval:

• To estimate area, partition the interval $\left[a,b\right]$ into $n$ subintervals.
• Width of each subinterval: $\mathrm{\Delta }x=\frac{b-a}{n}$.
3. 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.
4. 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: ${\int }_{a}^{b}f\left(x\right)\text{\hspace{0.17em}}dx={\mathrm{lim}}_{n\to \mathrm{\infty }}{\sum }_{i=1}^{n}f\left({x}_{i}\right)\mathrm{\Delta }x$.
5. Positive and Negative Areas:

• Positive values of $f\left(x\right)$ contribute to positive areas above the x-axis.
• Negative values of $f\left(x\right)$ contribute to negative areas below the x-axis.
6. Interpretation with Functions:

a. Above and Below the x-axis: For $f\left(x\right)\ge 0$ over $\left[a,b\right]$, the definite integral represents the area above the x-axis.

b. Negative Function Values: When $f\left(x\right)\le 0$ over $\left[a,b\right]$, the integral represents the negative of the area below the x-axis.

### 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.