Standard +0.8 This is a Further Maths volume of revolution question requiring integration of 1/(kΒ²+xΒ²), recognition of the arctan derivative, and solving Ο[arctan(1) - 0] = 1 for k. The integration step requires knowing a non-standard integral form, and the algebraic manipulation is straightforward but the setup requires careful handling of the constant k throughout.
4 The equation of a curve is \(\mathrm { y } = \frac { 1 } { \sqrt { \mathrm {~K} ^ { 2 } + \mathrm { x } ^ { 2 } } }\), where \(k\) is a positive constant. The region between the \(x\)-axis, the \(y\)-axis and the line \(x = k\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
Given that the volume of the solid of revolution formed is 1 unit \({ } ^ { 3 }\), find the exact value of \(k\).
4 The equation of a curve is $\mathrm { y } = \frac { 1 } { \sqrt { \mathrm {~K} ^ { 2 } + \mathrm { x } ^ { 2 } } }$, where $k$ is a positive constant. The region between the $x$-axis, the $y$-axis and the line $x = k$ is rotated through $2 \pi$ radians about the $x$-axis.
Given that the volume of the solid of revolution formed is 1 unit ${ } ^ { 3 }$, find the exact value of $k$.
\hfill \mbox{\textit{OCR MEI Further Pure Core 2024 Q4 [4]}}