Table of Contents

Approximating Double Integral

  1. What?
    The techniques discussed in the previous sections can be modified for use in the approximation of multiple integrals.

  2. Why?

  3. Comoposite Trapezoidal Rule for Double Integral:
    \(\ \ \iint_R f(x,y) \,dA \ = \ \int_{a}^{b} \ \big( \ \int_{c}^{d} \ \ f(x,y) \ \ dy \ \ \big) \ dx \ \ \\)

    \(\approx \dfrac{(b − a)(d − c)}{16} \bigg[f(a,c)+f(a,d) + f(b,c) + f(b,d)+\) \(\ \ \ \ \ \ \ \ 2\Big[f\big(\dfrac{a + b}{2} , c\big) + f\big(\dfrac{a + b}{2} , d\big) + f\big(a, \dfrac{c + d}{2}\big) + f\big(b, \dfrac{c + d}{2}\big)\Big] + 4f\big(\dfrac{a + b}{2}, \dfrac{c + d}{2}\big)\bigg]\)

  4. Comoposite Simpsons’ Rule for Double Integral:
    • Rule:
    • Error:
    • Derivation:

Gaussian Quadrature for Double Integral Approximation

  1. What?
    More efficient methods such as Gaussian quadrature, Romberg integration, or Adaptive quadrature can be incorporated in place of the Newton-Cotes formulas.
  2. Why?
    To reduce the number of functional evaluations.
  3. Example:

Non-Rectangular Regions

  1. What?
    Regions that don’t have a rectangular shape.
    Form:
    \[\ \int_{a}^{b} \bigg( \int_{c(x)}^{d(x)} f(x,y) dy \bigg) dx \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (4.42)\]
    or,
    \[\int_{c}^{d} \bigg( \int_{a(y)}^{b(y)} f(x,y) dx \bigg) dy \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (4.43)\]
  2. How?
    • We use Simpson’s Rule for Approximation.
    • Step Size:
      • x: \(\ \ h = \dfrac{b − a}{2}\)
      • y: \(\ \ k(x) = \dfrac{d(x) − c(x)}{2}\)
  3. Simpsons’ Rule for Non-Rect Regions:
  4. Simpsons’ Double Integral [Algorithm]:
  5. Gaussian Double Integral [Algorithm]:

Triple Integral Approximation

  1. What?
    • Triple integrals.
    • Form:
    \[\ \int_{a}^{b} \ \int_{c(x)}^{d(x)} \ \int_{\alpha(x)}^{\beta(x)} f(x,y) dz \ dy \ dx\]
  2. Gaussian Triple Integral [Algorithm]: