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

The curve here is \(y = (x-c)^2 + c^2\), and the white area is \(I(c)\). The size of the white area is given by the height of the red bar at the right.

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

What is always true about \((x-c)^2\) and \(c^2\)?

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

Can we complete the square here?

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

Notice we’re asked for the MAXIMUM value here…