Answer
a) 2$\sqrt[] 3$
b) $8$
Work Step by Step
a ) We need to find area:
$A(x)$ = $\frac{1}{2}(side)(side)(\sin\frac{\pi}{3})$=$\frac{1}{2}(2\sqrt {\sin x}(2\sqrt {\sin x})(\sin\frac{\pi}{3})$=$\sqrt 3 \sin x$
Then we find Volume :
$V$=$\int^b_aA(x) dx$ in here $a$=$0$ , $b$=$\pi$
$V$=$\int^b_aA(x) dx$=$\sqrt 3\int^\pi_0\sin x dx$=$[-\sqrt {\cos x}^\pi_0$=$\sqrt 3(1+1)$=$2\sqrt 3$
b) We need to find Area: $A(x)$ = $(side)^2$ = $(2\sqrt {\sin x})((2\sqrt {\sin x})$ =$4 \sin x$
Then we need to find Volume: $V$=$\int^b_aA(x)dx$. In here $a$=$0$, $b$=$\pi$
$V$=$\int^b_aA(x)dx$ = $\int^\pi_04 \sin x \ dx$ =$[-4 \cos x ]^\pi_0$ = $8$
