18M.1.SL.TZ2.S_2

pestleMathematicsAISLPaper 118M· sl-5-5-integration-introduction-areas-between-curve-and-x-axissource ↗

Let $f\left( x \right) = 6{x^2} - 3x$ . The graph of $f$ is shown in the following diagram.

Find $\int {\left( {6{x^2} - 3x} \right){\text{d}}x}$ .

[2]

Find the area of the region enclosed by the graph of $f$ , the x-axis and the lines x = 1 and x = 2 .

[4]

Markscheme / solution

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

$2{x^3} - \frac{{3{x^2}}}{2} + c\,\,\,\left( {{\text{accept}}\,\,\frac{{6{x^3}}}{3} - \frac{{3{x^2}}}{2} + c} \right)$ A1A1 N2

Notes: Award A1A0 for both correct terms if +c is omitted.
Award A1A0 for one correct term eg $2{x^3} + c$ .
Award A1A0 if both terms are correct, but candidate attempts further working to solve for c.

[2 marks]

substitution of limits or function (A1)

eg $\int_1^2 {f\left( x \right)} \,{\text{d}}x,\,\,\left[ {2{x^3} - \frac{{3{x^2}}}{2}} \right]_1^2$

substituting limits into their integrated function and subtracting (M1)

eg $\frac{{6 \times {2^3}}}{3} - \frac{{3 \times {2^2}}}{2} - \left( {\frac{{6 \times {1^3}}}{3} + \frac{{3 \times {1^2}}}{2}} \right)$

Note: Award M0 if substituted into original function.

correct working (A1)

eg $\frac{{6 \times 8}}{3} - \frac{{3 \times 4}}{2} - \frac{{6 \times 1}}{3} + \frac{{3 \times 1}}{2},\,\,\left( {16 - 6} \right) - \left( {2 - \frac{3}{2}} \right)$

$\frac{{19}}{2}$ A1 N3

[4 marks]

Examiners’ report

[N/A]