A-LevelMathematicsIntegration9709/22 · Feb/Mar 2020 · Q710 Marks
The diagram shows part of the curve with equation y = 4 sin²x + 8 sin x + 3, where x is measured in radians. The curve crosses the x-axis at the point A and the shaded region is bounded by the curve and the lines x = 0 and y = 0. [Figure 7.1]
- (a)Find the exact x-coordinate of A.[2]
- (b)Find the exact gradient of the curve at A.[3]
- (c)Find the exact area of the shaded region.[5]
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Mark scheme
Marking points from the official mark scheme (Feb/Mar 2020). The mark or mark type is shown in brackets.
- (a) Solve equation y = 0 to find value of x
- (a) Obtain 7/6 π
- (b) Attempt first derivative using chain rule OE
- (b) Obtain dy/dx = 8 sin x cosx + 8 cos x OE
- (b) Substitute value from part (a) to find gradient -2√3 Or exact equivalent
- (c) Express integrand in the form k₁ + k₂ cos 2x + k₃ sin x
- (c) Obtain correct 5-2 cos 2x + 8 sin x OE. Allow unsimplified
- (c) Integrate to obtain 5x-sin2x-8cosx
- (c) Apply limits 0 and their value from part (a) correctly
- (c) Obtain (35/6)π + (7/2)√3 + 8 or exact equivalent
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