Find the area of the region that lies outside the circle x2 + y2 = 1 but inside the circle x2 + y2 − 8y − 9 = 0.

Sure, let's solve this step by step.

Step 1: Identify the circles

The first circle is centered at the origin (0,0) with a radius of 1 (since r^2 = 1).

The second circle's equation can be rewritten in the standard form of a circle's equation, (x-h)^2 + (y-k)^2 = r^2, where (h,k) is the center and r is the radius. Completing the square for the y terms, we get:

x^2 + (y - 4)^2 = 16

So, the second circle is centered at (0,4) with a radius of 4.

Step 2: Find the area of the circles

The area of a circle is given by the formula πr^2. So, the area of the first circle is π(1)^2 = π, and the area of the second circle is π(4)^2 = 16π.

Step 3: Find the area of the region

The region that lies outside the first circle but inside the second circle is simply the area of the second circle minus the area of the first circle. So, the area of the region is 16π - π = 15π.