Mathematical Methods — Question 12

QCAA 2023, Paper 1 · 4 marks

Q12 · 2023 · Technology-freeSimple familiar4 marks

QUESTION 12 (4 marks)

The region bounded by the

x

-axis and the curve of

f(x)=x(4-x)

represents the plan of a garden bed. All measurements are in metres.

Estimate the area of the garden bed using sums of the form

\sum_i f(x_i)\,\delta x_i

, where

x_1=1

x_2=2

x_3=3

and

x_4=4

[1 mark]

Use a definite integral to determine the area of the garden bed.

[3 marks]

WORKED SOLUTION

4 marks

ANSWER

10\text{ m}^2

; b)

\dfrac{32}{3}\text{ m}^2=10\dfrac{2}{3}\text{ m}^2

Worked solution

a) Let

f(x)=x(4-x)

. Then

f(1)=3

f(2)=4

f(3)=3

f(4)=0

, so

\sum_i f(x_i)\delta x_i=(3\times1)+(4\times1)+(3\times1)+(0\times1)=10\text{ m}^2.

\text{Area}=\int_0^4x(4-x)\,dx=\int_0^4(-x^2+4x)\,dx.

=\left[-\frac13x^3+2x^2\right]_0^4=-\frac{64}{3}+32=\frac{32}{3}=10\frac23\text{ m}^2.

correctly calculates the described estimation

[1 mark]

correctly states the definite integral required

[1 mark]

determines the integration of the function

[1 mark]

calculates the area under the curve by integration

[1 mark]

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

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