Question about type 1 and type 2 regions

[Clara Chung]

Homework Statement

Homework Equations

The Attempt at a Solution

How is the regions in the given decomposition both type 1 and type 2 regions at the same time? Take the region at the upper right hand side as an example. It can't be neither type 1 and type 2 regions because the functions that are bounding the curve are not continuous. e.g. If I take the rightmost point as x=b, leftmost line as x=a, the lower function bounding y is not continuous.
Please help.

 

[LCKurtz]

I think you are confusing "continuous" with "two piece" or piecewise continuous functions. Taking the upper right piece as an example, it is a type 1 region with upper curve the corner curve and lower curve the two pieces given by an arc of a circle and a horizontal line. It would take two integrals to cover it as a dx integral. Similarly it is a type 2 region with the right curve being the corner curve and the left curve being two pieces, the vertical line segment and the arc of the inner circle. Consequently it would take two integrals to cover that region as a dy integral. But each segment is continuous even with the sharp corners.