1. **Problem Statement:** Compute the area enclosed by a straight line and a curve given the ordinates at intervals of 7.5 m using the Mid-ordinate rule, Trapezoidal rule, and Simpson's rule. 2. **Given Data:** Ordinates (heights) at equal intervals of 7.5 m: $$1.6, 2.9, 3.5, 8.3, 7.6, 9.9, 3.5, 6.9, 10.8, 5.6, 2.3, 11.2, 4.3$$ Number of ordinates, $n = 13$ Interval length, $h = 7.5$ m 3. **Formulas:** - Mid-ordinate rule (for $n$ ordinates): $$Area = h \times \sum_{i=1}^{n} y_i$$ - Trapezoidal rule (for $n$ ordinates): $$Area = \frac{h}{2} \left(y_1 + 2\sum_{i=2}^{n-1} y_i + y_n\right)$$ - Simpson's rule (requires odd number of ordinates, $n$ odd): $$Area = \frac{h}{3} \left(y_1 + 4\sum_{odd\ i} y_i + 2\sum_{even\ i} y_i + y_n\right)$$ 4. **Calculations:** **Mid-ordinate rule:** $$\sum y_i = 1.6 + 2.9 + 3.5 + 8.3 + 7.6 + 9.9 + 3.5 + 6.9 + 10.8 + 5.6 + 2.3 + 11.2 + 4.3 = 78.0$$ $$Area = 7.5 \times 78.0 = 585.0 \text{ m}^2$$ **Trapezoidal rule:** $$y_1 + y_n = 1.6 + 4.3 = 5.9$$ Sum of middle ordinates multiplied by 2: $$2 \times (2.9 + 3.5 + 8.3 + 7.6 + 9.9 + 3.5 + 6.9 + 10.8 + 5.6 + 2.3 + 11.2)$$ Calculate sum inside parentheses: $$2.9 + 3.5 + 8.3 + 7.6 + 9.9 + 3.5 + 6.9 + 10.8 + 5.6 + 2.3 + 11.2 = 72.5$$ Multiply by 2: $$2 \times 72.5 = 145.0$$ Total sum: $$5.9 + 145.0 = 150.9$$ Area: $$Area = \frac{7.5}{2} \times 150.9 = 3.75 \times 150.9 = 565.875 \text{ m}^2$$ **Simpson's rule:** Odd indices (excluding first and last): 3, 5, 7, 9, 11 Corresponding ordinates: $$y_3 = 3.5, y_5 = 7.6, y_7 = 3.5, y_9 = 10.8, y_{11} = 2.3$$ Sum odd indices: $$3.5 + 7.6 + 3.5 + 10.8 + 2.3 = 27.7$$ Even indices: 2, 4, 6, 8, 10, 12 Corresponding ordinates: $$y_2 = 2.9, y_4 = 8.3, y_6 = 9.9, y_8 = 6.9, y_{10} = 5.6, y_{12} = 11.2$$ Sum even indices: $$2.9 + 8.3 + 9.9 + 6.9 + 5.6 + 11.2 = 44.8$$ Apply Simpson's formula: $$Area = \frac{7.5}{3} \times \left(1.6 + 4 \times 27.7 + 2 \times 44.8 + 4.3\right)$$ Calculate inside parentheses: $$1.6 + 110.8 + 89.6 + 4.3 = 206.3$$ Area: $$Area = 2.5 \times 206.3 = 515.75 \text{ m}^2$$ 5. **Comments:** - Mid-ordinate rule overestimates area because it sums all ordinates directly. - Trapezoidal rule is more accurate by averaging adjacent ordinates. - Simpson's rule is generally the most accurate for smooth curves, but here it gives the smallest area. - Differences arise due to shape of curve and distribution of ordinates. Final answers: - Mid-ordinate rule area: $585.0$ m$^2$ - Trapezoidal rule area: $565.875$ m$^2$ - Simpson's rule area: $515.75$ m$^2$