Consider the tetrahedron $ \ T \ $ with vertices $ \ (1, 0, 0), (1, −1, 1), (1, 1, 1) \ \ and \ \ (0, 0, 1) \ $. How many regions must $ \ T \ $ be split into in order to integrate some function over T with the following variable orders ?
(i) $ dxdydz \ $
(ii) $ \ dxdzdy \ $
(iv) $ \ dydzdx \ $
Answer:
Suppose $ \ f(x,y,z) \ $ be any function over $ \ T \ $
(iv)
We have to integrate $ \ \iiint f(x,y,z) dxdydz \ $ over $ \ T \ $
Consider the plane containing the vertices $ \ (1,0,0) , \ (1,1,1) , \ (0,0,1) \ $ which is $ \ x-y+z=1 \ $
Thus the integration is
$ \int_{0}^{1} \int_{}^{} \int_{0}^{x+z-1} f(x,y,z) dxdzdy \ $
Thus there is only one region for (iv)
Am I right ?