it says the area outside the circle radius 2, and inside the cardoid.
your picture shows the inside of the circle shaded.
Further you say in the first octant but your picture shows the entire first quadrant included.
Clean things up please.
Remember that a full circle is $\displaystyle 2\pi$ radians (360 degrees).
In your posts above I think you are using $\displaystyle \pi$ radians for a full circle.
An octant is what you get when you divide a full circle into 8 parts.
I haven't gone through the notes you gave in detail, but the replies above from romsek and myself are correct - an octant is half of a quadrant i.e. an octant = 45 degrees, or $\displaystyle \frac{2\pi}{8}$ radians. I still think that from what you have posted you are using a full circle to be $\displaystyle \pi$ radians instead of $\displaystyle 2\pi$ radians.