The diagram shows the straight line x + y = 5 intersecting the curve y = 4/x at the points A (1, 4) and B (4, 1). Find, showing all necessary working, the volume obtained when the shaded region is rotated through 360° about the x-axis. [Figure 6.1]

Mark scheme

Marking points from the official mark scheme (May/June 2017). The mark or mark type is shown in brackets.

1. Vol = π ∫[(5-x)² dx - ∫(4/x)² dx]
2. [M1*] Use of volume formula at least once, condone omission of π and limits and dx.
3. [DM1] Subtracting volumes somewhere must be after squaring.
4. ∫(5-x)²dx = (5-x)³/(-3)
5. [B1 B1] Without ÷ (-1). B1 for ÷(-1)
6. (or 25x - 10x²/2 + x³/3)
7. [B2,1,0] -1 for each incorrect term
8. ∫(16/x²)dx = -16/x
9. [B1]
10. Use of limits 1 and 4 in an integrated expression and subtracted.
11. [DM1] Must have used “y²” at least once. Need to see values substituted.
12. → 9π or 28.3
13. [A1]