Question about type 1 and type 2 regions

  • #1
304
14

Homework Statement


62.png


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.
 

Attachments

Answers and Replies

  • #2
LCKurtz
Science Advisor
Homework Helper
Insights Author
Gold Member
9,556
767
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.
 
  • Like
Likes Clara Chung

Related Threads on Question about type 1 and type 2 regions

  • Last Post
Replies
0
Views
2K
  • Last Post
Replies
11
Views
2K
Replies
2
Views
722
Replies
1
Views
1K
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
1
Views
992
  • Last Post
Replies
3
Views
2K
Top