26.1. METHODS FOR DOUBLE INTEGRALS 525
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.
26.1.1 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
∫U ρ dA. In other words you integrate the density to get the mass. Now by letting
ρ depend on position, you can include the case where the material is not homogeneous.Here is an example.
Example 26.1.6 Let ρ (x,y) denote the density of the plane region determined by the curves13 x+ y = 2,x = 3y2, and x = 9y. Find the total mass if ρ (x,y) = y.
You need to first draw a picture of the region R. A rough sketch follows.
(3,1)
(9/2,1/2)
(0,0)
x = 3y2 (1/3)x+ y = 2
x = 9y
This region is in two pieces, one having the graph of x = 9y on the bottom and thegraph of x = 3y2 on the top and another piece having the graph of x = 9y on the bottom and