Let *R* be the region shown below.

Write an integral expression for the area of *R*, using vertical slices.

Hint

Use the sine and cosine functions to find the coordinates of the two points.

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