Show that
$$\sin x - \sin x \cos 2 x \approx 2 x ^ { 3 }$$
for small values of \(x\).
15
Hence, show that the area between the graph with equation
$$y = \sqrt { 8 ( \sin x - \sin x \cos 2 x ) }$$
the positive \(x\)-axis and the line \(x = 0.25\) can be approximated by
$$\text { Area } \approx 2 ^ { m } \times 5 ^ { n }$$
where \(m\) and \(n\) are integers to be found.
15
Explain why
$$\int _ { 6.3 } ^ { 6.4 } 2 x ^ { 3 } \mathrm {~d} x$$
is not a suitable approximation for
$$\int _ { 6.3 } ^ { 6.4 } ( \sin x - \sin x \cos 2 x ) d x$$
Question 15 continues on the next page
15
(ii) Explain how
$$\int _ { 6.3 } ^ { 6.4 } ( \sin x - \sin x \cos 2 x ) d x$$
may be approximated by
$$\int _ { a } ^ { b } 2 x ^ { 3 } \mathrm {~d} x$$
for suitable values of \(a\) and \(b\).
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