26.1. METHODS FOR DOUBLE INTEGRALS 523

of values of the function times areas of little rectangles. This is why it is entirely reasonableto expect the iterated integrals in two different orders to be equal. It is also why the iteratedintegral is approximating something which we call the Riemannn integral.

For another more precise explanation for equality of iterated integrals in the case wherethe function is continuous, see Problem 7 on Page 238. For the whole story, see the chapteron the Lebesgue integral.

Definition 26.1.2 Let R be a bounded region in the xy plane and let f be a boundedfunction defined on R. We say f is Riemannn integrable if there exists a number, denotedby∫

R f dA and called the Riemannn integral such that if ε > 0 is given, then wheneverone imposes a sufficiently fine mesh enclosing R and considers the finitely many rectangleswhich intersect R, numbered as {Qi}m

i=1 and a point (xi,yi) ∈ Qi, it follows that∣∣∣∣∣∫

Rf dA−∑

if (xi,yi)area(Qi)

∣∣∣∣∣< ε

It is∫

R f dA which is of interest. The iterated integral should always be considered as atool for computing this number. When this is kept in mind, things become less confusing.Also, it is helpful to consider

∫R f dA as a kind of a glorified sum. It means to take the value

of f at a point and multiply by a little chunk of area dA and then add these together, hencethe integral sign which is really just an elongated symbol for a sum.

The careful explanation of these ideas is contained later in a special chapter devoted tothe theory of the integral. I have presented there the Lebesgue integral because it is mucheasier to understand and use although it is more abstract.

Example 26.1.3 Let f (x,y) = x2y+ yx for (x,y) ∈ R where R is the triangular region de-fined to be in the first quadrant, below the line y = x and to the left of the line x = 4. Find∫

R f dA.

x

y

4

R

From the above discussion,∫R

f dA =∫ 4

0

∫ x

0

(x2y+ yx

)dydx

The reason for this is that x goes from 0 to 4 and for each fixed x between 0 and 4, y goesfrom 0 to the slanted line, y = x, the function being defined to be 0 for larger y. Thus y goesfrom 0 to x. This explains the inside integral. Now

∫ x0(x2y+ yx

)dy = 1

2 x4 + 12 x3 and so∫

Rf dA =

∫ 4

0

(12

x4 +12

x3)

dx =672

5.

What of integration in a different order? Lets put the integral with respect to y on theoutside and the integral with respect to x on the inside. Then∫

Rf dA =

∫ 4

0

∫ 4

y

(x2y+ yx

)dxdy