How does your answer to \(\displaystyle{\int_a^b \sqrt{1-x^2} \, dx}\) relate to the following diagram?

Look carefully at the diagram above.

Do you know any information that you can add to the diagram?

Are there lengths and areas that you can work out?

Further suggestions can be found by working through the questions below:

Find \(p\) in terms of \(a\).

As this is a unit circle the hypotenuse of the triangle is \(1\). Therefore \(p = \sqrt{1-a^2}\).

Find \(\theta\) in terms of \(a\).

As we know all three sides of the triangle there are multiple ways we could write the angle, but the simplest way would be to use \(\cos\theta = \frac{a}{1}\) which means \(\theta = \arccos{a}.\)

Find \(\phi\) and the shaded area in terms of \(a\) and \(b\).

It will be helpful to find the area of the sector using radians rather than degrees.

We already know one angle marked in the digram below, \(\arccos a.\) The other can be found in the same way and is \(\arccos b.\)
Therefore \(\phi = \arccos a - \arccos b\).
The area of a sector \(= \frac{1}{2}r^2\theta,\) and \(r = 1\) , so the area is \(\frac{1}{2}(\arccos a - \arccos b).\)