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

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

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