13.2 Double Integrals over General Regions
Suppose the region is not rectangular. Integrate

 
2
0
cos
0
e sin drd
Def: An xy region (or xy plane) is called a type I region if any
vertical strip (in the y direction) always has the same upper
and lower boundaries and the region can be described by a set
of inequalities a ≤ x ≤ b and h(x) ≤ y ≤ g(x).
Def: An xy region (or xy plane) is called a type II region if any
horizontal strip (in the x direction) always has the same left
and right boundaries and the region can be described by a set
of inequalities c ≤ y ≤ d and h(y) ≤ x ≤ g(y).
b g(x )

If type II use  
If type I use
a h(x )
c g(y )
d

f (x, y) dydx
h(y )
f (x, y) dxdy
Ex. Suppose R is bounded by y = x2 and x + y = 6.

Suppose R is the triangular region bounded by (1, 3), (2, 1)
and (4, 4).
Sketch the regions of integration and reverse the order of
integration.
6 3
Ex.
  f (x, y)
0 1
dydx

Ex.
4
y
0
0

f (x, y) dxdy

e ln x
Ex.


0 0
e y dydx
1
Ex.
 
y 1
1  y 1
f (x, y) dxdy

Ex. Integrate



0
x
 
sin y
dydx
y
Do: Sketch the region and reverse the order of integration for
1 y
1.

0 y2
2
2.



1
y2
y
f (x, y) dxdy
f (x, y) dxdy
Ex. Find the volume of the solid whose base is the region in
the xy-plane that is bounded by the parabola y = 4 – x2 and the
line y = 3x, while the top of the solid is bounded by the plane
z = x + 4.