Answer

Following the hint, let's find the coordinates of the points. Since we're looking at the unit circle, the point at angle θ has coordinates (cos θ, sin θ).

We can fill in the coordinates of the point at angle immediately:

To find the angle for the other point, we subtract from :

We get

,

so the coordinates of the second point are

.

If we're using vertical slices we need to split the region *R* into two pieces at , because, for , the upper bound of *R* is a line, and for , the upper bound of *R* is the circle.

Let's find the area of the left piece of *R*.

For the upper bound of *R* is a line that goes through the origin and has slope

The equation of this line is .

The lower bound of *R* is the line *y* = *x*. This means the height of the slice at location *x* (for ) is

The area of the slice is

Since *x* runs from 0 to in the left piece of *R*, the area of the left piece of *R* is

.

Now for the right piece of *R*.

For the upper bound of *R* is the equation of the circle, *x*^{2} + *y*^{2} = 1 or

.

The lower bound of *R* is still the line *y* = *x*.

The height of the slice at position *x* (for ) is

The area of the slice is

Since *x* runs from to in the right piece of *R*, the area of the right piece of *R* is

Putting these integrals together, the area of *R* is