Chapter 19

The Riemannn Integral On Rn

19.1 Methods For Double IntegralsThis chapter is on the Riemannn integral for a function of n variables. It begins by in-troducing the basic concepts and applications of the integral. The general considerationsincluding the definition of the integral and proofs of theorems are left till later. These arevery difficult topics and are likely better considered in the context of the Lebesgue integral.Consider the following region which is labeled R.

R

a b

y = t(x)

y = b(x)

y

x

We will consider the following iterated integral which makes sense for any continuousfunction f (x,y) . ∫ b

a

∫ t(x)

b(x)f (x,y)dydx

It means just exactly what the notation suggests it does. You fix x and then you do theinside integral ∫ t(x)

b(x)f (x,y)dy

This yields a function of x which will end up being continuous. You then do∫ b

a dx to thiscontinuous function.

What was it about the above region which made it possible to set up such an iteratedintegral? It was just this: You have a curve on the top y = t (x) , and a curve on the bottomy = b(x) for x ∈ [a,b]. You could have set up a similar iterated integral if you had a regionin which there was a curve on the left and a curve on the right for y in some interval. Hereis an example.

Example 19.1.1 Suppose t (x) = 4−x2,b(x) = 0 and a =−2,b = 2. Compute the iteratedintegral described above for f (x,y) = xy+ y.

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