15. numerical integration

15.1 integration as the limit of a sum

• split rectangles appropriately with equal width,

• create x and y table (DON’T FORGET THE 0 IF INTEGRATION WANTS IT),

• total area: width of rectangle x (y1 + y2 + … + yn)

  • INCREASING graph/function:

    • (UB) ignore first x/y value, take the last,

    • (LB) take first x/y value, ignore the last.

  • DECREASING graph/function:

    • (UB) take first x/y value, ignore the last,

    • (LB) ignore first x/y value, take the last.

15.2 the trapezium rule

• area of a trapezium = (a + b) h / 2

• (FB) integral of f(x) from b to a:

  • h/2 (y₀ + y_n + 2 (y₁ + y₂ + … + y_n-1))

    • AKA h/2 times (first + last + twice the rest)

  • because, the middle y values are counted twice,

  • h is the equal distance between trapeziums or otherwise b-a / n (number of equal intervals)

• ordinates are the lines splitting the trapeziums, ordinates - 1 are the intervals.

• there’s a table function in your calculator btw.

• trapezium rule UNDERESTIMATES when CONCAVE,

• trapezium rule OVERESTIMATES when CONVEX.