Area and Volume Computation
Core concepts
Area computation uses the trapezoidal rule, Simpson's rule, the mid-ordinate rule, and the coordinate (shoelace) method. Understand when each applies: Simpson's rule needs an even number of equal intervals (odd number of ordinates) and assumes a parabolic boundary; the trapezoidal rule works with any number of equal intervals.
Volume computation uses the trapezoidal (average end area) formula, the prismoidal formula, and the prismoidal correction. Know the prismoidal formula: V = h/6 (A₁ + 4Aₘ + A₂). For earthwork, understand formation level, side slopes, and the mid-section method for a cutting or embankment.
Traps in questions
- Simpson's rule requires an odd number of ordinates; using it with an even number is a common trap.
- The prismoidal formula is more accurate than average end area; the correction is average end area minus prismoidal.
- For a trapezoidal channel or embankment, side slopes (e.g., 1:2) affect the cross-sectional area.
- Coordinate method: area = ½ |Σ(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ)|; sign indicates direction, not magnitude.
Worked example / rule of thumb
Ordinates 2, 4, 6, 8, 10 m at 10 m intervals (5 ordinates, 4 intervals). Trapezoidal: A = d[(o₁+oₙ)/2 + sum of middles] = 10[(2+10)/2 + 4+6+8] = 10[6+18] = 240 m². Simpson's: A = d/3 [(o₁+oₙ) + 4(sum of even ordinates) + 2(sum of odd ordinates)] = 10/3[(2+10) + 4(4+8) + 2(6)] = 10/3[12+48+12] = 240 m². Rule: always count intervals, not points.