The diagram shows part of the curve with equation $$\(y=x^{3}-2 b x^{2}+b^{2} x\)$$ and the line $$\(O A\)$$, where $$\(A\)$$ is the maximum point on the curve. The $$\(x\)$$-coordinate of $$\(A\)$$ is $$\(a\)$$ and the curve has a minimum point at $$\((b, 0)\)$$, where $$\(a\)$$ and $$\(b\)$$ are positive constants. Show that the area of the shaded region between the line and the curve is $$\(k a^{4}\)$$, where $$\(k\)$$ is a fraction to be found. .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... ....................................................................................................................................................
Exam No：9709_s20_qp_13 Year：2020 Question No：11(b)
Answer:
Area under curve \(=\int\left(x^{3}-6 a x^{2}+9 a^{2} x\right) \mathrm{d} x\)
\(\frac{x^{4}}{4}-2 a x^{3}+\frac{9 a^{2} x^{2}}{2}\)
\(\frac{a^{4}}{4}-2 a^{4}+\frac{9 a^{4}}{2}\left(=\frac{11 a^{4}}{4}\right)\)
(M1 for applying limits \(0 \rightarrow a\) )
When \(x=a, y=a^{3}-6 a^{3}+9 a^{3}=4 a^{3}\)
Area under line \(=\frac{1}{2} a \times\) their \(4 a^{3}\)
Shaded area \(=\frac{11 a^{4}}{4}-2 a^{4}=\frac{3}{4} a^{4}\)
\(\frac{x^{4}}{4}-2 a x^{3}+\frac{9 a^{2} x^{2}}{2}\)
\(\frac{a^{4}}{4}-2 a^{4}+\frac{9 a^{4}}{2}\left(=\frac{11 a^{4}}{4}\right)\)
(M1 for applying limits \(0 \rightarrow a\) )
When \(x=a, y=a^{3}-6 a^{3}+9 a^{3}=4 a^{3}\)
Area under line \(=\frac{1}{2} a \times\) their \(4 a^{3}\)
Shaded area \(=\frac{11 a^{4}}{4}-2 a^{4}=\frac{3}{4} a^{4}\)
Knowledge points:
1.8.3 evaluate definite integrals (Including simple cases of ‘improper’ integrals, such as)
1.8.4.1 the area of a region bounded by a curve and lines parallel to the axes, or between a curve and a line or between two curves
Solution:
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