Question

(5) Rewrite the integral Z 1 0 Z x 0 Z √ 1−x2 0 z dz dy dx in the order dy dx dz, and evaluate it. Hint: You will need a reasonably good idea of what the solid over which the integration is being done looks like to have any hope of getting correct limits for the iterated integral in the new order of integration.

Answer 1

Looks like the integral is supposed to read

$\displaystyle\int_0^1\int_0^x\int_0^{\sqrt{1-x^2}}z\,\mathrm dz\,\mathrm dy\,\mathrm dx$

The same integral in the required order is

$\displaystyle\int_0^1\int_0^{\sqrt{1-z^2}}\int_0^xz\,\mathrm dy\,\mathrm dx\,\mathrm dz$