A-LevelMathematicsNumerical methods9709/22 · Feb/Mar 2024 · Q47 Marks
The diagram shows the curve with equation y = √1+e⁰·⁵ˣ. The shaded region is bounded by the curve and the straight lines x = 0, x = 6 and y = 0. [Diagram]
- (a)Use the trapezium rule with three intervals to find an approximation to the area of the shaded region. Give your answer correct to 3 significant figures.[3]
- (b)The shaded region is rotated completely about the x-axis. Find the exact volume of the solid produced.[4]
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Mark scheme
Marking points from the official mark scheme (Feb/Mar 2024). The mark or mark type is shown in brackets.
- [B1] (a) Use y-values √2, √1+e, √1+e², √1+e³ [Or decimal equivalents]
- [M1] (a) Use correct formula, or equivalent, with h=2
- [A1] (a) Obtain 15.7 [Or greater accuracy.]
- [B1] (b) State or imply that volume is π∫(1+e⁰·⁵ˣ) dx [Implied if π appears only at the end.]
- [M1] (b) Integrate to obtain form k₁x+k₂e⁰·⁵ˣ [Where k₁k₂ ≠0 with or without π.]
- [A1] (b) Obtain π(x + 2e⁰·⁵ˣ) or x + 2e⁰·⁵ˣ
- [A1] (b) Obtain 4π + 2πε³ [Or exact equivalent.]
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