contestada

Find the area of the region bounded by the courses y = sin^-1(x/6), y = 0, and x = 6 obtained by integrating with respect to y. Your work must include the definite integral and the antiderivative.

Respuesta :

Given

The equation is given

[tex]y=sin^{-1}(\frac{x}{6})[/tex]

y=0 and x=6.

Required

To determine the area of region bounded by courses of the equation given.

Explanation[tex]\begin{gathered} y=sin^{-1}(\frac{x}{6})\Rightarrow\Rightarrow siny=\frac{x}{6} \\ 6siny=x \end{gathered}[/tex][tex]x=6,y=0[/tex]

Therefore

[tex]\begin{gathered} 6siny=6 \\ siny=1 \\ y=\frac{\pi}{2} \end{gathered}[/tex][tex]\begin{gathered} 6siny=0 \\ siny=0 \\ y=0,\pi \end{gathered}[/tex]

Now integrate the equation.

[tex]\int_0^{\frac{\pi}{2}}(6-6siny)dy=(6y+6cosy)_0^{\frac{\pi}{2}}[/tex][tex](\frac{6\pi}{2}+6cos\frac{\pi}{2})-(6\times0+6cos0)=3\pi-6[/tex]Answer

Hence the area of region bounded by the courses is

[tex]3\pi-6[/tex]

Otras preguntas