The integral _0^1 ! ! _ x ^ 1 e^ y^3

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Calculus of change and accumulation · Calculus of change and accumulation

Mathematical Methods for Physics, Ch Multiple integrals and change of variables

Problem

The integral∫01 ⁣ ⁣∫x1ey3 dy dx\int_0^1\!\!\int_{\sqrt x}^{1} e^{y^3}\,\mathrm{d}y\,\mathrm{d}xcannot be done in the order written, because ey3e^{y^3} has no elementary antiderivative. Sketch the region, reverse the order of integration, and evaluate the integral.

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