2. Evaluate the double integral iint_{R} x y d x d y where R is the region in the first quadrant bounded by y=3 x^{2}, y=4-x^{2}.

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Accepted Answer

Step:1 ：Find the intersection points of the two curves (y=3x^2) and (y=4-x^2). These points will give us the limits of integration for (x).Step:2 ：The intersection points are ((-1, 3)) and ((1, 3)).Step:3 ：Sort the (x)-values of the intersection points in ascending order to get the limits of integration for (x). The sorted (x)-values are ([-1, 1]).Step:4 ：Integrate the function (xy) with respect to (x) first and then with respect to (y).Step:5 ：The double integral evaluates to zero.Step:6 ：Final Answer: (boxed{0})

Answer

Step: 1 ：Find the intersection points of the two curves (y=3x^2) and (y=4-x^2). These points will give us the limits of integration for (x).Step: 2 ：The intersection points are ((-1, 3)) and ((1, 3)).Step: 3 ：Sort the (x)-values of the intersection points in ascending order to get the limits of integration for (x). The sorted (x)-values are ([-1, 1]).Step: 4 ：Integrate the function (xy) with respect to (x) first and then with respect to (y).Step: 5 ：The double integral evaluates to zero.Step: 6 ：Final Answer: (boxed{0})

2. Evaluate the double integral $\iint_{R} x y d x d y$ where $R$ is the region in the first quadrant bounded by $y=3 x^{2}, y=4-x^{2}$.

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