Find the region before you choose the radii

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

Total marks: 10

Solution

(i)

Stationary points

The curve $C$ has equation $y=3-\dfrac{10}{x^2-2x+4}$ . Without a calculator, determine the exact coordinates of its stationary point.

Hint

Differentiate. The denominator is a quadratic, so use the chain rule on $10(x^2-2x+4)^{-1}$ .
Set $\frac{dy}{dx}=0$ and solve.
Or complete the square first: the denominator is $(x-1)^2+3$ .

Answer · (i)

(1,-1/3)

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

Stationary points

Differentiate

Solve $\frac{dy}{dx}=0$

Check it is a minimum

Sketch first, then choose radii

Differentiate

Solve dydx=0\frac{dy}{dx}=0dxdy​=0

Check it is a minimum

Sketch first, then choose radii

Solve $\frac{dy}{dx}=0$