Integration Area Afe6B2

### **Area of a Right Triangle** 1. **Problem Statement:** Calculate the area of a right triangle with base and height both equal to 4 units. 2. **Formula Used:** Area = \(\frac{1}{2} \times \text{base} \times \text{height}\) 3. **Calculation:** Area = \(\frac{1}{2} \times 4 \times 4 = 8\) square units. --- ### **Calculating Area Under Curves - Introduction to Integration** 1. **Problem Statement:** Find the area under the curve \(y = x^2\) from \(x=0\) to \(x=2\). 2. **Concept:** Complex shapes like curves do not have simple area formulas. Integration breaks the area into many thin rectangles. 3. **Approximation:** Using rectangles of width 0.2, number of rectangles = \(\frac{2}{0.2} = 10\). 4. **Height of Each Rectangle:** Height = \(y = x^2\) evaluated at each rectangle's \(x\)-value. 5. **Area of Each Rectangle:** Area = \(\text{base} \times \text{height} = 0.2 \times x^2\). 6. **Summation:** Sum areas of all rectangles to approximate total area. --- ### **Defining Integration** 1. **Concept:** As the rectangle width \(\Delta x\) approaches zero, the sum becomes an integral. 2. **Integral Form:** \(\int_0^2 x^2 \, dx\) represents the exact area under \(y=x^2\) from 0 to 2. 3. **Integration Formula:** For \(x^n\), \(\int x^n dx = \frac{x^{n+1}}{n+1} + C\). --- ### **Calculating Exact Area Using Integration** 1. **Apply Formula:** \(\int_0^2 x^2 dx = \left[ \frac{x^3}{3} \right]_0^2 = \frac{8}{3} \approx 2.67\) 2. **Comparison:** This is close to the approximation using 10 rectangles. --- ### **Integration for Right Triangle Using Curve \(y=x\)** 1. **Curve Representation:** Right triangle with base and height 4 can be represented by \(y = x\) from 0 to 4. 2. **Integral:** \(\int_0^4 x \, dx = \left[ \frac{x^2}{2} \right]_0^4 = \frac{16}{2} = 8\) 3. **Result:** Matches the area calculated by the simple triangle formula. --- ### **Real-life Applications of Integration** - Engineers calculate material quantities for curved structures. - Economists find total income over time from earnings curves. - Physicists compute distances from velocity-time graphs. --- **Summary:** Integration generalizes area calculation from simple shapes to complex curves by summing infinitesimally small rectangles. Understanding this bridges geometry and calculus effectively.