EXM.2.AHL.TZ0.12

pestleMathematicsAIHLPaper 2EXM· sl-5-8-trapezoid-rulesource ↗

The curve $y = f\left( x \right)$ is shown in the graph, for $0 \leqslant x \leqslant 10$ .

The curve $y = f\left( x \right)$ passes through the following points.

It is required to find the area bounded by the curve, the $x$ -axis, the $y$ -axis and the line $x = 10$ .

One possible model for the curve $y = f\left( x \right)$ is a cubic function.

Use the trapezoidal rule to find an estimate for the area.

[3]

Use all the coordinates in the table to find the equation of the least squares cubic regression curve.

[3]

b.i.

Write down the coefficient of determination.

[1]

b.ii.

Write down an expression for the area enclosed by the cubic regression curve, the $x$ -axis, the $y$ -axis and the line $x = 10$ .

[1]

c.i.

Find the value of this area.

[2]

c.ii.

Markscheme / solution

Area = $\frac{2}{2}\left( {2 + 2\left( {4.5 + 4.2 + 3.3 + 4.5} \right) + 8} \right)$ M1A1

Area = 43 A1

[3 marks]

$y = 0.0389{x^3} - 0.534{x^2} + 2.06x + 2.06$ M1A2

[3 marks]

b.i.

${R^2} = 0.991$ A1

[1 mark]

b.ii.

Area = $\int\limits_0^{10} {y\,dx}$ A1

[1 mark]

c.i.

42.5 A2

[2 marks]

c.ii.

Examiners’ report

[N/A]

b.i.

[N/A]

b.ii.

[N/A]

c.i.

[N/A]

c.ii.