The diagram shows the curve with equation x = y² + 1. The points A (5, 2) and B (2, -1) lie on the curve. [Figure 11.1]

Mark scheme

Marking points from the official mark scheme (Feb/Mar 2023). The mark or mark type is shown in brackets.

1. [M1] Gradient of AB = (2-(-1)) / (5-2) Expect 1, must be from Δy / Δx.
2. [A1] Equation of AB is y-2=1(x-5) or y+1=1(x-2) OE. Expect y = x-3.
3. [M1] For curve: [π]∫x²dy=[π]∫(y²+1)²dy = [π]∫(y⁴+2y²+1)dy For curve: Attempt to square y²+1 and attempt integration. Subtracting curve equation from line equation before squaring is M0. Integration before squaring M0.
4. [A2, 1, 0] [π](y⁵/5 + 2y³/3 + y)
5. [M1] For line: [π]∫(y+3)²dy = [π]∫(y²+6y+9)dy For line: Attempt to square their y+3 and attempt integration.
6. [A2, 1, 0] [π](y³/3 + 3y² + 9y) or [π](y+3)³/3 Not available for incorrect line equations.
7. [DM1] Apply limits -1→2 to either integral providing they have been awarded M1. Expect 15.6 [π] and/or 39[π]. Some evidence of substitution of both - 1 and 2 must be seen. Dependent on at least one of the first 2 M1 marks.
8. [DM1] Appropriate subtraction. Dependent on at least one of the first 2 M1 marks.
9. [A1] Volume = [π](39 – 15.6) = 23.4π or 117/5π or awrt 73.5[1327]