Arc length and surface area — AS & A Level Further Mathematics (9231) questions by topic
Parametric arc-length and surface-of-revolution integrals define this small topic: showing that a hyperbolic-parametrised curve's length reduces to an integral involving cosh 2t, and using a substitution to derive the surface area generated when y = e^x is rotated about the x-axis.
Show, find and deduce are the command words across 6 questions from 2023 to 2024 papers, worth from 7 to 10 marks, so each response is typically a single sustained derivation rather than several short parts. With no examiner-report extracts recorded here, working through each substitution and simplification step with instant marking is the way to confirm an integral has been reduced to the exact required form before submitting the final answer.
May–June 2024, Paper 23
The curve C has parametric equations x = cosh t, y = sinh t, for 0 < t <= 3/5. The length of C is denoted by s. Show that s = the integral from t = 0 to t = 3/5 of sqrt(cosh 2t)…
Answer this question and get it marked →October–November 2024, Paper 21
A curve has equation y = e^x for ln(4/3) <= x <= ln(12/5). The area of the surface generated when the curve is rotated through 2pi radians about the x-axis is denoted by A. Use…
Answer this question and get it marked →October–November 2024, Paper 22
The curve C has parametric equations x = (1/2)e^(2t) - (1/3)t^3 - 1/2, y = 2 e^t (t - 1), for 0 <= t <= 1. Find the exact length of C.
Answer this question and get it marked →October–November 2024, Paper 23
A curve has equation y = e^x for ln(4/3) <= x <= ln(12/5). The area of the surface generated when the curve is rotated through 2pi radians about the x-axis is denoted by A. Use…
Answer this question and get it marked →October–November 2023, Paper 21
The curve C has parametric equations x = (2/3) t^(3/2) - 2 t^(1/2), y = 2t + 5, for 0 < t <= 3. Find the exact length of C.
Answer this question and get it marked →October–November 2023, Paper 23
The curve C has parametric equations x = (2/3) t^(3/2) - 2 t^(1/2) and y = 2t + 5, for 0 < t <= 3. Find the exact length of C.
Answer this question and get it marked →Other AS & A Level Further Mathematics topics
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