Q168 · Practice questionTechnology-activeComplex familiar6 marks
QUESTION 168 (6 marks)
The region lies below and above the -axis for . The marked points are at .
a)[2 marks]
Determine the exact area of .
b)[2 marks]
Use Simpson's rule with the five marked points to approximate . Give the approximation as an exact fraction.
c)[2 marks]
State whether the approximation is an overestimate or underestimate, and calculate the percentage error relative to the exact area to four decimal places.
WORKED SOLUTION
6 marksPractice marking scheme
ANSWER
(a) . (b) . (c) Underestimate by approximately .
Worked solution
(a) Since an antiderivative is ,
(b) The five ordinates are . There are four subintervals with width , so
(c) , so the approximation is an underestimate. Its percentage error relative to the true area is
The exact area is used as the denominator. Five points give four subintervals, not five. Angles for the integral are interpreted in radians.
Equivalent mathematically justified methods accepted.
Exact answers unless a decimal accuracy is specified.
Part a: Set up and integrate the area using arctangent.
Part a: Obtain arctan(2).
Part b: Use width 1/2 and weights 1,4,2,4,1.
Part b: Obtain 431/390.
Part c: Identify the underestimate.
Part c: Use the exact-area denominator and obtain 0.1825 percent.
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