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Three of these four hold for the double integral. The exception is

  • A.it reduces to a repeated single integral
  • B.its value is unchanged by a valid change of variables with the Jacobian included
  • C.it gives the area of the region when the integrand is one
  • D.its value depends on the order in which the two integrations are performed

Answer: D

For a continuous integrand over a reasonable region the value is the same in either order, which is Fubini's theorem; what changes is the difficulty. Reduction to a repeated integral, invariance under a properly weighted change of variables, and the area interpretation are all correct

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