17N.1.SL.TZ0.S_8

pestleMathematicsAASLPaper 117N· sl-5-3-differentiating-polynomials-n-e-zsource ↗

Let $f(x) = {x^2} - x$ , for $x \in \mathbb{R}$ . The following diagram shows part of the graph of $f$ .

N17/5/MATME/SP1/ENG/TZ0/08

The graph of $f$ crosses the $x$ -axis at the origin and at the point ${\text{P}}(1,{\text{ }}0)$ .

The line $L$ intersects the graph of $f$ at another point Q, as shown in the following diagram.

N17/5/MATME/SP1/ENG/TZ0/08.c.d

Find the area of the region enclosed by the graph of $f$ and the line $L$ .

Markscheme / solution

valid approach (M1)

eg $\,\,\,\,\,$ $\int {L - f,{\text{ }}\int_{ - 1}^1 {(1 - {x^2}){\text{d}}x} }$ , splitting area into triangles and integrals

correct integration (A1)(A1)

eg $\,\,\,\,\,$ $\left[ {x - \frac{{{x^3}}}{3}} \right]_{ - 1}^1,{\text{ }} - \frac{{{x^3}}}{3} - \frac{{{x^2}}}{2} + \frac{{{x^2}}}{2} + x$

substituting their limits into their integrated function and subtracting (in any order) (M1)

eg $\,\,\,\,\,$ $1 - \frac{1}{3} - \left( { - 1 - \frac{{ - 1}}{3}} \right)$

Note: Award M0 for substituting into original or differentiated function.

area $= \frac{4}{3}$ A2 N3

[6 marks]

Examiners’ report

[N/A]