Review question

# Can we find the area under a parabola? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R6067

## Suggestion

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…