A city council is planning to create an artificial lake of depth 2.5 m. Three si

Question

A city council is planning to create an artificial lake of depth 2.5 m. Three sides of the lake (western, southern and eastern sides) are to be straight lines at right angles, however, the northern end of the lake is bounded by a winding road. Landscape architects took the following measurements: Estimate, using the composite Trapezoidal rule for n = 3 subintervals, how many cubic metres of dirt have to be excavated for the lake. Estimate, using the composite Simpson rule for n = 3 subintervals, how many cubic metres of dirt have to be excavated for the lake.

Explanation / Answer

xe[0,120] i) n=3 ==> 120/3=40 ==> h=40 x=0 ==> f(x) =250 x=40 ==> f(x) =70 x=80 ==> f(x) = 48 x=120 ==> f(x) =28 composite trapezoidal rule for n=3 integral_{x=0}^{120} = h( 0.5* f(0) + f(40) + f(80) +0.5* f(120) ) = 40 ( 0.5* (250) + (70) + (48) +0.5* (28) ) = 10280 ii) simpson's rule n=3 ==> 120/3=40 ==> h=40 x=0 ==> f(x) =250 x=40 ==> f(x) =70 x=80 ==> f(x) = 48 x=120 ==> f(x) =28 simpson's rul rule for n=3 integral_{x=0}^{120} = (3 h/ 8) ( f(0) + 3*f(40) + 3*f(80) +0.5* f(120) ) = (120/8) ( 250 + 3*(70) + 3*(48) + 28 ) = 9480

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