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Question 27.11: Sketch the region R over which we would evaluate the integra......

Sketch the region R over which we would evaluate the integral

∫y=0y=1∫x=0x=2−2yf(x,y) dx dy\int_{y=0}^{y=1}\int_{x=0}^{x=2-2y}f(x,y)\,{\mathrm{d}}x\,\mathrm{d}y
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First consider the outer integral. The restriction on y means that interest can be confined to the horizontal strip 0 ≤ y ≤ 1. Then examine the inner integral. The lower limit on x means that we need only consider values of x greater than or equal to 0. The upper x limit depends upon the value of y. If y = 0 this upper limit is x = 2 − 2y = 2. If y = 1 the upper limit is x = 2−2y = 0. At any other intermediate value of y we can calculate the corresponding upper x limit. This upper limit will lie on the straight line x = 2−2y. With this information the region of integration can be sketched. The region is shown in Figure 27.12 and is seen to be triangular.

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