19.1. METHODS FOR DOUBLE INTEGRALS 353
agonizing. A good rule of thumb is that if what you do is easy it will be wrong. Thereare no shortcuts! There are no quick fixes which require no thought! Pain and sufferingis inevitable and you must not expect it to be otherwise. Always draw a picture and thenbegin agonizing over the correct limits. Even when you are careful you will make lots ofmistakes until you get used to the process.
Sometimes an integral can be evaluated in one order but not in another.
Example 19.1.5 For R as shown below, find∫
R sin(y2)
dA.
x
8
4
R
Setting this up to have the integral with respect to y on the inside yields∫ 4
0
∫ 8
2xsin(y2) dydx.
Unfortunately, there is no antiderivative in terms of elementary functions for sin(y2)
sothere is an immediate problem in evaluating the inside integral. It doesn’t work out so thenext step is to do the integration in another order and see if some progress can be made.This yields ∫ 8
0
∫ y/2
0sin(y2) dxdy =
∫ 8
0
y2
sin(y2) dy
and∫ 8
0y2 sin
(y2)
dy =− 14 cos64+ 1
4 which you can verify by making the substitution, u =
y2. Thus ∫R
sin(y2) dy =−1
4cos64+
14.
This illustrates an important idea. The integral∫
R sin(y2)
dA is defined as a number.It is the unique number between all the upper sums and all the lower sums. Finding it isanother matter. In this case it was possible to find it using one order of integration but notthe other. The iterated integral in this other order also is defined as a number but it cannot befound directly without interchanging the order of integration. Of course sometimes nothingyou try will work out.
19.1.1 Density And MassConsider a two dimensional material. Of course there is no such thing but a flat platemight be modeled as one. The density ρ is a function of position and is defined as follows.Consider a small chunk of area dA located at the point whose Cartesian coordinates are(x,y). Then the mass of this small chunk of material is given by ρ (x,y) dA. Thus if thematerial occupies a region in two dimensional space U , the total mass of this materialwould be ∫
Uρ dA