1. Chapter III. Numerical Integration for Ship Forms Review of CVEN 302

2. Data of Ship forms Discrete data (Line drawings, stations, water plane etc) Evenly distributed (most times)

3. Methods of Numerical Integration Trapezoidal rule (linear) Sinpson’s 1/3 rule (quadratic) Simpson’s 3/8 rule (cubic) Multiple applications Tchebycheff’s (similar to Gauss Quadrature) rule -applied to a continues function

4.  f n (x) can be linear  f n (x) can be quadratic

5.  f n (x) can also be cubic or other higher-order polynomials

6. Trapezoidal Rule (single Application) Linear approximation x0x0 x1x1 x f(x)f(x) L(x)

7. Multiple Applications of Trapezoidal Rule x0x0 x1x1 x f(x)f(x) x2x2 hhx3x3 hhx4x4

8. Simpson’s 1/3-Rule (single application) Approximate the function by a parabola x0x0 x1x1 x f(x)f(x) x2x2 hh L(x)L(x)

9. Multiple Applications of Simpson’s 1/3 Rule  Applicable only if the number of segments is even

10. Multiple Applications of Simpson’s 1/3 Rule n must be even

11. Simpson’s 3/8-Rule (single application)  Approximate by a cubic polynomial x0x0 x1x1 x f(x) x2x2 hh L(x) x3x3 h

12. Tchebycheff’s rule See Table 4.3 at p58 Positions of ordinates (stations) depending on how many ordinates are used Odd # of ordinates is preferred