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 (x-h)^2 + (y-k)^2 = r^2 by completing the square. This gives us (x-0)^2 + (y-4)^2 = 4^2, so it's centered at (0,4) with a radius of 4.

Step 2: Visualize the circles If you draw these circles, you'll see that the smaller circle is entirely contained within the larger one. The region we're interested in is the area of the larger circle excluding the area of the smaller circle.

Step 3: Calculate the areas The area of a circle is given by πr^2. So, the area of the smaller circle is π(1^2) = π, and the area of the larger circle is π(4^2) = 16π.

Step 4: Subtract the areas The area of the region we're interested in is the area of the larger circle minus the area of the smaller circle. So, the area of the region is 16π - π = 15π.