Integration by substitution: examples

Evaluate the integral

 ʃʃB ( x2 + y2 ) dA ,

where B is an ellipse

   ( 5 / 8 )x2 + ( 3 √ 3 / 8 )xy + y2 = 1,

rotated through ( π / 3 ). This is the rotation:

 F( u, v ) = |cos( π / 3 ) -sin( π / 3 )| |u|
             |sin( π / 3 )  cos( π / 3 )| |v|
           = |( 1 / 2 ) ( -√3 / 2 )| |u|    
             |( √3 / 2 )  ( 1 / 2 )| |v|

Thus in the rotated system we have

 x = ( 1 / 2 )u - ( √3 / 2 )v
 y = ( √3 / 2 )u + ( 1 / 2 )v

Now

 ʃʃB ( x2 + y2 ) dA = ʃʃR [( ( 1 / 2 )u - ( 3 / 2 )v )2 + ( ( 3 / 2 )u + ( 1 / 2)v)2)] dA.

By recalling that

 F' = |( 1 / 2 ) ( -√3 / 2 )|
      |( √3 / 2 )  ( 1 / 2 )|,

we find that

 det F' = ( 1 / 2 ) ( 1 / 2 ) + ( √3 / 2 ) ( √3 / 2 ) = ( 1 / 4 ) + ( 3 / 4 ) = 1.

Now we want to find the formula describing the boundary of R. For the boundary ∂B of the region B we have, after we substitute:

 ( 5 / 8 )( ( 1 / 2 )u - ( √3 / 2 )v )2 + ( 3 √ 3 / 8 )( ( 1 / 2 )u - ( √3 / 2 )v )( ( √3 / 2 )u + ( 1 / 2 )v ) 
                       + ( ( √3 / 2 )u + ( 1 / 2 )v )2 = 1.

Simplifying we get

 …u2 + …v2 = 1.     (iterated integral)