Find the region before you choose the radii

Curve C: y=3−10/(x2−2x+4)y=3-10/(x^2-2x+4).

(i) Without a calculator, determine its exact stationary point. marks

(ii) Sketch C, stating clearly any asymptotes. marks

R is bounded by C, 3y+16x=153y+16x=15 and the yy-axis.

(iii) Find its exact area. marks

(iv) Find the volume when R is rotated 2π2\pi radians about the yy-axis. marks

[2021/EJC/Prelim/II/4]

Total marks: 10

Solution

(i)

Stationary points

The curve CC has equation y=3−10x2−2x+4y=3-\dfrac{10}{x^2-2x+4}. Without a calculator, determine the exact coordinates of its stationary point.

Hint

  1. Differentiate. The denominator is a quadratic, so use the chain rule on 10(x2−2x+4)−110(x^2-2x+4)^{-1}.

  2. Set dydx=0\frac{dy}{dx}=0 and solve.

  3. Or complete the square first: the denominator is (x−1)2+3(x-1)^2+3.

Answer · (i)

Minimum (1,−1/3)(1,-1/3).

(ii)

Curve sketching

Sketch CC, stating clearly any asymptotes.

Explore this idea

Hint

  1. What happens to the fraction 10(x−1)2+3\frac{10}{(x-1)^2+3} as x→±∞x\to\pm\infty?

  2. Find the yy-intercept (x=0x=0) and the xx-intercepts (y=0y=0).

  3. Plot the minimum from (i) and use the symmetry about x=1x=1.

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

Answer · (ii)

Asymptote y=3y=3; intercepts (0,1/2),(1±1/3,0)(0,1/2),(1\pm1/\sqrt3,0).

Explore the idea

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

Area between curves

RR is the region bounded by CC, the line 3y+16x=153y+16x=15 and the yy-axis. Find its exact area.

Explore this idea

Hint

  1. Write the line as y=5−16x3y=5-\frac{16x}{3}. It meets CC where? Try the minimum point.

  2. Which is on top between x=0x=0 and that point?

  3. Integrate (line − curve). Expect an arctangent.

Answer · (iii)

Area =5π/(33)−2/3=5\pi/(3\sqrt3)-2/3.

Explore the idea

The line sits above the curve until they touch at the minimum

Each strip has height (line − curve). The strips shrink to nothing at x = 1, where the line meets the curve.

Number of stripsn = 4
Try an example

Where does one strip’s area come from?

1 of 4

Click a strip in the graph to inspect it. The darker rectangle is this strip. Its red marks use the left edge at x* = 0.

upperlowerwidth Δx
Zoomed-in rectangle: width and height are scaled independently. The sample sets its upper and lower values across the whole strip.
Sample x*
0
Width Δx
0.25
Upper value
5
Lower value
0.5
Height: upper − lower
4.5
Strip area
1.125

0.25 ×(5 − 0.5)≈ 1.125

Add this contribution to those of the other 3 strips to get the sum below. Displayed values are rounded.

Strip width
0.250
1.000 ÷ 4
Sum of strips
2.91231
Exact, 5π/(3√3) − ⅔
2.35633
Error
+0.55598
23.60% too big

How the sum improves as n grows

  • Top of R: y = 5 − 16x/3
  • Bottom of R: the curve C
  • Rectangles of height (line − curve)
  • Exact area ≈ 2.3563
Key idea

R is a tall wedge that closes up at (1,−13)(1,-\frac13). The strips converge on 5π33−23≈2.356\frac{5\pi}{3\sqrt3}-\frac23\approx2.356.

Ideas to try
  1. Switch to midpoints and compare with the left-edge rule at n = 6.
  2. Look at the right end. Why do the strips become so thin?
(iv)

Volume of revolution

Find the volume generated when RR is rotated through 2π2\pi radians about the yy-axis.

Explore this idea

Hint

  1. Rotating about the yy-axis, slice horizontally, so everything must be a function of yy.

  2. The outer boundary is the line. For low yy the inner boundary is the curve; for high yy it is the yy-axis itself.

  3. The outer solid is a cone. Subtract the hollow left by the curve.

Answer · (iv)

Volume =2π[−7/9+5ln⁡(3/4)+5π/(33)]≈5.07=2\pi[-7/9+5\ln(3/4)+5\pi/(3\sqrt3)]\approx5.07 cubic units.

Explore the idea

A cone with a curved hollow

Sweep R round the y-axis: the line makes a cone, the curve carves a hollow into its base. Slice it into washers or shells.

Open the interactive view to explore this geometry in 3D.

Takeaway

Sketch first, then choose radii

The region touches the axis only for y≥12y\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.