The diagram shows the curve $$$y=\sin 2 x \cos ^{2} x$$$ for $$$0 \leqslant x \leqslant \frac{1}{2} \pi$$$, and its maximum point $$$M$$$. Using the substitution $$$u=\sin x$$$, find the exact area of the region bounded by the curve and the  $$$x$$$-axis. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mathematics

IGCSE&ALevel

CAIE

Exam No：9709_m21_qp_32 Year：2021 Question No：10(a)

Answer:

State or imply $\mathrm{d} u=\cos x \mathrm{~d} x$
Using double angle formula for $\sin 2 x$ and Pythagoras, express integral in terms of $u$
and $\mathrm{d} u$.
Obtain integral $\int 2\left(u-u^{3}\right) \mathrm{d} u$
Use limits $u=0$ and $u=1$ in an integral of the form $a u^{2}+b u^{4}$, where $a b \neq 0$
Obtain answer $\frac{1}{2}$

Knowledge points:

3.3.2.3 the formulae for sin 2A and tan 2A

3.5.6 use a given substitution to simplify and evaluate either a definite or an indefinite integral.

Answer:

Knowledge points:

Solution:

Related Question