Let \[I(c)=\int_0^1 ((x-c)^2 +c^2)dx\] where \(c\) is a real number.

  1. Sketch \(y=(x-1)^2+1\) for the values \(-1\le x\le 3\) and show on your graph the area represented by the integral \(I(1)\).

  2. Without explicitly calculating \(I(c)\), explain why \(I(c) \ge 0\) for any value of \(c\).

  3. Calculate \(I(c)\).

  4. What is the minimum value of \(I(c)\) (as \(c\) varies)?

  5. What is the maximum value of \(I(\sin \theta)\) as \(\theta\) varies?