Q211 · Practice questionTechnology-freeComplex unfamiliar5 marks
QUESTION 211 (5 marks)
For on [0,2], use ordinates at x=0,1,2. For comparison, the composite trapezoidal rule here is .
a)[3 marks]
Find the exact integral, Simpson approximation and composite trapezoidal approximation.
b)[2 marks]
A student claims trapezoidal integration is exact for every cubic. Give a cubic counterexample on the same interval.
WORKED SOLUTION
5 marksPractice marking scheme
ANSWER
(a) All three equal . (b) : integral , trapezoidal value .
Worked solution
(a) The odd term around x=1 integrates to zero; the constant gives 4. Ordinates are 1,2,3. With step 1, Simpson gives and trapezoidal gives .
(b) The exact integral of x cubed is . Ordinates 0,1,8 give trapezoidal value . Thus the claim fails; the original agreement arose from symmetry.
Equivalent mathematically justified methods accepted.
Exact answers unless a decimal accuracy is specified.
Part a: Evaluate the exact integral.
Part a: Calculate Simpson value.
Part a: Calculate the trapezoidal value.
Part b: Choose a cubic and evaluate its integral.
Part b: Calculate a different trapezoidal result to refute the claim.
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