The diagram shows part of the curve with equation $$\[ y=4 \sin ^{2} x+8 \sin x+3, \]$$ where $$\(x\)$$ is measured in radians. The curve crosses the $$\(x\)$$-axis at the point $$\(A\)$$ and the shaded region is bounded by the curve and the lines $$\(x=0\)$$ and $$\(y=0\)$$. Find the exact area of the shaded region. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................
Exam No：9709_m20_qp_22 Year：2020 Question No：7(c)
Answer:
Express integrand in the form \(k_{1}+k_{2} \cos 2 x+k_{3} \sin x\)
Obtain correct \(5-2 \cos 2 x+8 \sin x\)
Integrate to obtain \(5 x-\sin 2 x-8 \cos x\)
Apply limits 0 and their value from part (a) correctly
Obtain \(\frac{35}{6} \pi+\frac{7}{2} \sqrt{3}+8\) or exact equivalent
Obtain correct \(5-2 \cos 2 x+8 \sin x\)
Integrate to obtain \(5 x-\sin 2 x-8 \cos x\)
Apply limits 0 and their value from part (a) correctly
Obtain \(\frac{35}{6} \pi+\frac{7}{2} \sqrt{3}+8\) or exact equivalent
Knowledge points:
2.5.1 extend the idea of 'reverse differentiation' to include the integration of (Knowledge of the general method of integration by substitution is not required.)
2.5.2 use trigonometrical relationships in carrying out integration
Solution:
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