7. A circle \(C\) with centre \(O\) and radius \(r\) has cartesian equation \(x ^ { 2 } + y ^ { 2 } = r ^ { 2 }\) where \(r\) is a constant.
- Show that \(1 + \left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) ^ { 2 } = \frac { r ^ { 2 } } { r ^ { 2 } - x ^ { 2 } }\)
- Show that the surface area of the sphere generated by rotating \(C\) through \(\pi\) radians about the \(x\)-axis is \(4 \pi r ^ { 2 }\).
- Write down the length of the arc of the curve \(y = \sqrt { } \left( 1 - x ^ { 2 } \right)\) from \(x = 0\) to \(x = 1\)