[Solved] Change the order of integration in \(\mathop{\int }

Changing the order of integration | MIT 18.02SC Multivariable Calculus, Fall 2010
Changing the order of integration | MIT 18.02SC Multivariable Calculus, Fall 2010

Question

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Change the order of integration in

\(\mathop{\int }_{0}^{a}\mathop{\int }_{y}^{a}\frac{x}{{{x}^{2}}+{{y}^{2}}}dx~dy\)

Answer (Detailed Solution Below)

Detailed Solution

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Concept:

In a double integral with variable limits, the change of order of integration can sometimes make the evaluation easy.

Calculation:

Since the limit of y is constant hence, we can say that a horizontal strip was taken for integration now for change of order we will take vertical strip as shown in the following diagram.

∴ \(\mathop{\int }_{0}^{a}\mathop{\int }_{0}^{x}\frac{x}{{{x}^{2}}+{{y}^{2}}}dy~dx\) is the correct solution.

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