Techniques of Integration

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Revised

22 Mar 2023

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Factoring

When the bounds of integration are independent (when they do not contain variables), then multiple integrals may be factored.

rectangle \( \begin{aligned} \int_c^d\int_a^bx^2y^3\,dx\,dy &=\int_c^dy^3\,dy\int_a^bx^2\,dx \end{aligned} \)

we are integrating wrt x first, so we can factor out all y terms

since the bounds don’t depend on y, dy can also be factored out

yields two easy integrals multiplied together

\( \begin{aligned} \int_1^2\int_0^3x^2y^3\,dx\,dy &=\int_1^2y^3\,dy\int_0^3x^2\,dx \end{aligned} \)

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