A-LevelMathematicsIntegration9709/32 · Oct/Nov 2022 · Q88 Marks
The diagram shows part of the curve y = sin √x. This part of the curve intersects the x-axis at the point where x = a. [Figure 8.1]
- (a)State the exact value of a.[1]
- (b)Using the substitution u = √x, find the exact area of the shaded region in the first quadrant bounded by this part of the curve and the x-axis.[7]
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Mark scheme
Marking points from the official mark scheme (Oct/Nov 2022). The mark or mark type is shown in brackets.
- [B1] (a) State (a =) π²
- [B1] (b) State or imply dx = 2u du or equivalent
- [M1] (b) Substitute for x and dx throughout the integral
- [A1] (b) Obtain ∫ 2u sin u du
- [*M1] (b) Commence integration of ∫ ku sin u du by parts and reach ±ku cos u ± ∫ k cos u du
- [A1] (b) Obtain integral -ku cos u + k sin u
- [DM1] (b) Substitute limits u = 0 and u = √their a, a ≠ 0, a in radians or x = 0 and their a in the complete integral
- [A1] (b) Obtain answer 2π
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