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    Integrals, Jacobian, Integration, and Integrals

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    1. The double integral (see attached) is determined in the domain D shown in the figure below. The domain boundaries are determined by the following functions:
    1) y1(x) = x;
    2) y2(x) = (9-x^2)^(1/2);
    3) y3(x) = -x;
    4) y4(x) = (1-x^2)^(1/2)

    (a) Simplify the integrant, reduce the integral to the polar coordinates and indicate the limits of integration
    (b) Calculate the integral

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    Solution Preview

    The intergrand is:

    f (x, y) = {x^2 sin^2 (y) + exp [ln(x^2)] cos^2 (y) + y^2} exp (x^2 + y^2)

    We start with the identity exp [ln(a)] = ...

    Solution Summary

    The solution shows in detail how to convert an integral from Cartesian to polar coordinates, calculate the Jacobian, and determine the new domain of integration and evaluates the integral.