2 A curve has equation \(\mathrm { y } = \sqrt { 1 + \mathrm { x } ^ { 2 } }\), for \(0 \leqslant x \leqslant 1\), where both the \(x\) - and \(y\)-units are in cm. The area of the surface generated when this curve is rotated fully about the \(x\)-axis is \(A \mathrm {~cm} ^ { 2 }\).
- Show that \(\mathrm { A } = 2 \pi \int _ { 0 } ^ { 1 } \sqrt { 1 + \mathrm { kx } ^ { 2 } } \mathrm { dx }\) for some integer \(k\) to be determined.
A small component for a car is produced in the shape of this surface. The curved surface area of the component must be \(8 \mathrm {~cm} ^ { 2 }\), accurate to within one percent. The engineering process produces such components with a curved surface area accurate to within one half of one percent.
- Determine whether all components produced will be suitable for use in the car.