Standard +0.3 This is a standard area-between-curves problem requiring finding intersection points by solving a quadratic equation, then integrating the difference of two functions. While it involves multiple steps (finding intersections, setting up the integral, integrating a quadratic and linear function), all techniques are routine for P1 level with no conceptual challenges or novel insights required.
\includegraphics{figure_9}
The diagram shows the curve \(y = (x - 2)^2\) and the line \(y + 2x = 7\), which intersect at points \(A\) and \(B\). Find the area of the shaded region. [8]
Elimination of \(y \rightarrow x^2 - 2x - 3 = 0 \rightarrow A(-1, 9)\) and \(B(3, 1)\)
M1, DM1A1
\(y\) (or \(x\)) removed completely. Soln of quadratic. Both points correct.
Area under line \(= \frac{1}{2} × 4 × 10\) or \(∫[7x - x^2]\) from \(-1\) to \(3\).
M1
Uses any valid method – integration or area of trapezium etc.
Area under curve \(= \left[\frac{(x-2)^3}{3}\right]\)
M1
Any attempt at integration.
or \(\left[\frac{x^3}{3} - 2x^2 + 4x\right]\) from \(-1\) to \(3\)
M1
Any attempt at integration.
\(\rightarrow 10\frac{2}{3}\)
A1 [8]
Correct integration in either form. Correct use of limits in an integral. co.
[ok to use \(∫(y_1 - y_2)dx\) – marks the same]
$y = (x-2)^2$ and $y + 2x = 7$
Elimination of $y \rightarrow x^2 - 2x - 3 = 0 \rightarrow A(-1, 9)$ and $B(3, 1)$ | M1, DM1A1 | $y$ (or $x$) removed completely. Soln of quadratic. Both points correct.
Area under line $= \frac{1}{2} × 4 × 10$ or $∫[7x - x^2]$ from $-1$ to $3$. | M1 | Uses any valid method – integration or area of trapezium etc.
Area under curve $= \left[\frac{(x-2)^3}{3}\right]$ | M1 | Any attempt at integration.
or $\left[\frac{x^3}{3} - 2x^2 + 4x\right]$ from $-1$ to $3$ | M1 | Any attempt at integration.
$\rightarrow 10\frac{2}{3}$ | A1 [8] | Correct integration in either form. Correct use of limits in an integral. co.
[ok to use $∫(y_1 - y_2)dx$ – marks the same]
\includegraphics{figure_9}
The diagram shows the curve $y = (x - 2)^2$ and the line $y + 2x = 7$, which intersect at points $A$ and $B$. Find the area of the shaded region. [8]
\hfill \mbox{\textit{CAIE P1 2010 Q9 [8]}}