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

If this part uses an earlier result, revisit (ii) Curve sketching or open all working together.

(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

Open interactive exploration

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?
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.