Find the region before you choose the radii

y=3-10/(x^2-2x+4)

Total marks: 10

Solution

If this part uses an earlier result, revisit (i) Stationary points or open all working together.

(ii)

Curve sketching

Sketch $C$ , stating clearly any asymptotes.

Explore this idea

Hint

What happens to the fraction $\frac{10}{(x-1)^2+3}$ as $x\to\pm\infty$ ?
Find the $y$ -intercept ( $x=0$ ) and the $x$ -intercepts ( $y=0$ ).
Plot the minimum from (i) and use the symmetry about $x=1$ .

Rational trough and cone subtractionScroll across to see the full diagram.

Answer · (ii)

y=3

Explore the idea

Open interactive exploration

Switch on the features, one at a time

Each switch adds something an examiner expects to see, and explains how you would find it.

Switch on a feature to add it to the sketch and see how it is found. A full-mark sketch shows all of them.

Graph of C
Horizontal asymptote y = 3
Line of symmetry x = 1
Intercepts with the axes

Ideas to try

Switch on “Symmetry” and then “Minimum”. Why does one tell you the other?
Which two features would you have missed without completing the square?

Takeaway

Sketch first, then choose radii

The region touches the axis only for $y\ge\frac12$ . Below that the curve moves away from the axis, which is what creates the hollow. Seeing that on a sketch is what tells you whether you need one integral or a subtraction.

Find the region before you choose the radii

Solution

Curve sketching

Asymptote

Intercepts

Draw it

Switch on the features, one at a time

Sketch first, then choose radii