Lecture Notes on Geometry and Calculus

Introduction to Geometry and Area Calculation

Right Triangle Area Calculation
- Formula: (\text{Area} = \frac{\text{Base} \times \text{Height}}{2})
- Example: Right triangle with base = 4 units, height = 4 units
  - Calculation: (4 \times 4 / 2 = 8) square units
- A triangle is half of a rectangle, hence the formula.

Complex Shapes
- Not all shapes are simple like triangles/rectangles (e.g., curves).
- Real-world problems involve complex shapes without straight edges (e.g., U-shaped valley).
Calculus and Integration
- Used for finding areas under curves.
- Integration allows approximation of areas by dividing shapes into thin rectangles.

Approximating Curved Areas
- Example: Parabola represented by (y = x^2).
- Divide into thin vertical slices (rectangles) with small width (dx).
- Approximation improves as (dx) approaches zero (infinitely small width).
Area Calculation Using Integration
- For parabola (x^2):
  - Area of each rectangle = (x^2 \times dx)
  - Sum these areas using integration
- Integration notation: (\int x^2 , dx)
- Formula for integration: (\int x^n , dx = \frac{x^{n+1}}{n+1})
Example Calculation
- Area under parabola from (x=0) to (x=2)
- Calculate: ([\frac{x^3}{3}]_{0}^{2} = \frac{8}{3} \approx 2.67)
- Comparison: Result is close to calculation using 10 rectangles._

Right Triangle Using Integration
- Represented by curve (y = x) from (x = 0) to (x = 4).
- Integration: (\int x , dx = \frac{x^2}{2})
- Result: (\frac{16}{2} = 8), same as using basic geometric formula.

Integration is a powerful tool for solving real-world problems involving complex shapes.
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