ABSTRACT
Let the range of integration be divided into n equal intervals each of width d , such that nd D ba, i.e. d D b a
n The ordinates are labelled y1, y2, y3, . . . ynC1 as
shown. An approximation to the area under the curve may
be determined by joining the tops of the ordinates by straight lines. Each interval is thus a trapezium, and since the area of a trapezium is given by:
area D 1 2
(sum of parallel sides) (perpendicular distance between them) then∫ b
a y dx ³ 1
2 y1 C y2 d C 12 y2 C y3 d
C 1 2 y3 C y4 d C Ð Ð Ð 12 yn C ynC1 d
³ d [
1 2 y1 C y2 C y3 C y4 C Ð Ð Ð C yn
C 1 2 ynC1
i.e. the trapezoidal rule states:
(1)
Problem 1. (a) Use integration to evaluate, correct to 3 decimal places,
2p x
dx (b) Use the trapezoidal rule with 4 intervals to evaluate the integral in part (a), correct to 3 decimal places
2p
dx D ∫ 3
2x
D
1 2
C 1
D [
D 4 [px ]31 D 4 [p3 p1] D 2.928, correct to 3 decimal
places.
