2024 VJC J2 CA2 Q5

Given that \(\mathrm{f}\) is a continuous function, explain, with the aid of a sketch, why the value of

\({\displaystyle \lim_{n\to\infty} \frac{\pi}{4n}\left\{\mathrm{f}\left(\frac{\pi}{4n}\right)+\mathrm{f}\left(\frac{2\pi}{4n}\right)+\ldots+\mathrm{f}\left(\frac{n\pi}{4n}\right)\right\}}\)

is \(\displaystyle\displaystyle \displaystyle\int_{0}^{\frac{\pi }{4}}{\mathrm{f}\left( x \right)\mathrm{d}x}\).[2]

Hence evaluate \({\displaystyle \lim_{n\to\infty} \frac{\pi}{4n}\left\{\sec^2\left(\frac{\pi}{4n}\right)+\sec^2\left(\frac{2\pi}{4n}\right)+\ldots+2\right\}.}\)[2]

The curve with equation \(y={{x}^{3}}+1\) is transformed by a translation of \(1\) unit in the negative \(y\)-direction. State the equation of the new curve in the form \(y=\mathrm{f}\left( x \right)\).[1]

The region \(R\) is bounded by the curve with equation \(y={{x}^{3}}+1\) and the lines \(y=1\) and \(x=b\), where \(b\) is a positive constant.

Given that the volume of the solid formed when \(R\) is rotated through \(2\pi \) radians about the line \(y=1\) is the same as the volume of the solid formed when \(R\) is rotated through \(2\pi \) radians about the \(y\)-axis, find the value of \(b\).[4]