2020 NYJC CT2 P1 Q8

By using the substitution \(x=a\sin \theta \), show \(\displaystyle\displaystyle \displaystyle\int_{0}^{a}{\,\sqrt{1-\frac{{{x}^{2}}}{{{a}^{2}}}}\,\mathrm{d}x=\frac{a\pi }{4}}\).[3]

The finite region, \(R\), in the first quadrant is bounded by the curve \(\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\) and the line \(\frac{x}{a}+\frac{y}{b}=1\), where \(a>b>0\). Find the exact value of the area of \(R\).[3]

By writing down the equation of the resulting curve when \(\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\) is translated \(b\) units in the direction of the negative \(y\)- axis, show that the volume of the region \(R\) when it is rotated though \(2\pi \) radians about the line \(y=b\) is given by \(\frac{\pi a{{b}^{2}}}{6}(3\pi -8)\).[4]