For this question, how to evaluate the integral by changing the order of integration? Also, how to sketch the region of integration? I really get stuck.
2026-04-01 15:04:02.1775055842
Multiple integrals: Double integrals
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Hint: Integrate first with respect to $y$, where $y$ goes from $0$ to $x^3$. We end up needing to integrate $2\pi x^2 \sin(\pi x^2)$, an easy substitution.
To get the region, draw $x=y^{1/3}$, or equivalently $y=x^3$. We are integrating over the region in the first quadrant which is below this curve and to the left of the line $x=1$.