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EXAMPLE 3 Evaluate اور ان ہی کی R er+y² dy dx, where R is the semicircular region bounded by the x-axis and the curve y = V1 - x² (Figure 15.27). Solution In Cartesian coordinates, the integral in question is a nonelementary integral and there is no direct way to integrate e²+² with respect to either x or y. Yet this integral and others like it are important in mathematics-in statistics, for example and we need
to evaluate it. Polar coordinates make this possible. Substituting x = r cos and y = r sin and replacing dy dx by r dr de give [[ ²² dy dx = ["['e'r dr do = ["[T« e²+y² R TT = "12 (e − 1) do = 7 (e − 1). The r in the r dr do is what allowed us to integrate e. Without it, we would have been unable to find an antiderivative for the first (innermost) iterated integral.