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Question

# (a) sketch the region of integration, (b) switch the order of integration, and (c) use a computer algebra system to show that both orders yield the same value. ∫_0^2∫_√4-x²^4-x²/4 xy / x²+y²+1 dy dx

Solution

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Expression given is written in both sequence below

\begin{align*}\Rightarrow I= \int _{0}^2\int _{4-x^2}^{4- x^2/4} \dfrac{xy}{x^2 + y^2 +1} dx dy &= \int _{0}^2 \int _{\sqrt{4-y^2}}^{2} \dfrac{xy}{x^2 + y^2 +1} dy dx \end{align*}

Graph of the area bounded is

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