Use y as the monotone coordinate · source reference

Printed question

C: $x=3t^2$, $y=t^3+3t$, where $t$ is real. A line l cuts C at two points P and Q with parameters $t=1$ and $t=-1/2$, respectively, as shown. Find the area bounded by C and l to 3 significant figures. [5]

[2011/TPJC/Prelim/I/1]

Printed working

At P, $t=1,x=3,y=4$. At Q, $t=-1/2,x=3/4,y=-13/8$. $y=t^3+3t\Rightarrow dy/dt=3t^2+3$.

$$\text{Area}=\frac12(3/4+3)(4+13/8)-\int_{-13/8}^4x\,dy=\frac{675}{64}-\int_{-1/2}^1(3t^2)(3t^2+3)dt=5.32\quad\text{units}^2.$$

Teacher notes

Area of red region = area of trapezium $=\frac12h(d_1+d_2)$. The source graph labels h between Q and P levels, $d_1$ at P and $d_2$ at Q; subtract the blue area between C and the $y$-axis.

Printed answer key

$5.32$ units$^2$.

Source diagrams in native form

Mathematical review

The named P–Q region has exact area $1701/320=5.315625$, rounding to $5.32$. The line is $y=(5/2)x-7/2$. Besides the named parameters, substitution gives $(t-1)(2t+1)(t-7)=0/2$: there is a third intersection at $t=7$ outside the drawn P–Q region. The stated two-point description is incomplete; the native source diagram intentionally presents the requested P–Q region and this additional intersection is recorded here.