check the picture below.
let's check where the quadratic and x = 6 meet, bear in mind we're integrating by "y", so
[tex]\bf y=sin^{-1}\left( \frac{x}{6} \right)\implies sin(y)=\cfrac{x}{6}\implies \boxed{6sin(y)=x}
\\\\\\
\begin{cases}
6sin(y)=x\\
x=6\\
y=0\\
x=0\leftarrow y-axis
\end{cases}\\\\
-------------------------------\\\\
6sin(y)=6\implies sin(y)=\cfrac{6}{6}\implies sin(y)=1\implies \measuredangle \boxed{y=\frac{\pi }{2}}
\\\\\\
6sin(y)=0\implies sin(y)=0\implies \to \boxed{\measuredangle y=0\ ,\ \pi }[/tex]
[tex]\bf -------------------------------\\\\
\displaystyle \int\limits_{0}^{\frac{\pi }{2}}\ [6-6sin(y)]dy\implies \left. \cfrac{}{} 3y^2+6cos(y) \right]_{0}^{\frac{\pi }{2}}
\\\\\\
\left[ \cfrac{3\pi 2}{4}+0 \right]-[0+6]\implies \cfrac{3\pi^2}{4}-6\approx 1.4022033[/tex]
