Mathematical Methods — Question 11

QCAA 2024, Paper 2 · 4 marks

Q11 · 2024 · Technology-activeSimple familiar4 marks

QUESTION 11 (4 marks)

State the trapezoidal rule and use it with six strips to determine an approximate value of the definite integral for the curve of

f(x)=4(x-3)^2

from

x=0

x=3

. Show all substitutions made into the rule.

WORKED SOLUTION

4 marks

ANSWER

36.5

units

^2

Worked solution

The strip width is

w=\frac{b-a}{n}=\frac{3-0}{6}=\frac12.

The trapezoidal rule is

A\approx\frac{w}{2}[f(x_0)+2f(x_1)+2f(x_2)+\cdots+f(x_n)].

Therefore

A\approx\frac{1/2}{2}[36+2(25)+2(16)+2(9)+2(4)+2(1)+0]=36.5\text{ units}^2.

correctly determines the rectangle width

[1 mark]

correctly states the trapezoidal rule

[1 mark]

substitutes appropriate values into the trapezoidal rule

[1 mark]

determines the approximate value of the definite integral

[1 mark]

One QCAA sample method typeset for web; criterion wording is adapted from the official marking guide.

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