Challenging +1.2 This is a standard surface area of revolution question for parametric curves requiring the formula A = 2π∫y√((dx/dt)² + (dy/dt)²)dt. The derivatives are straightforward (dx/dt = 2t - 2/t, dy/dt = 4), and the resulting integral simplifies nicely to give an exact answer. While it's a Further Maths topic with multiple steps, it's a direct application of a bookwork formula with no conceptual challenges or novel problem-solving required.
A curve is given by \(x = t^2 - 2\ln t\), \(y = 4t\) for \(t > 0\). When the arc of the curve between the points where \(t = 1\) and \(t = 4\) is rotated through \(2\pi\) radians about the \(x\)-axis, a surface of revolution is formed with surface area \(A\).
Given that \(A = k\pi\), where \(k\) is an integer, write down an integral which gives \(A\) and find the value of \(k\). [4]
A curve is given by $x = t^2 - 2\ln t$, $y = 4t$ for $t > 0$. When the arc of the curve between the points where $t = 1$ and $t = 4$ is rotated through $2\pi$ radians about the $x$-axis, a surface of revolution is formed with surface area $A$.
Given that $A = k\pi$, where $k$ is an integer, write down an integral which gives $A$ and find the value of $k$. [4]
\hfill \mbox{\textit{OCR Further Additional Pure 2017 Q1 [4]}}