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.

  1. 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 \(c^2\)?

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

Can we complete the square here?

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

Notice we’re asked for the MAXIMUM value here…