This is because according to the integral limit of the repeated integral given in the question, the region D of this double integral can be explained as follows.

D={(r，θ)|0≤θ≤∏/2，0≤r≤√sin2θ}

And r=√sin2θ is Bernoulli lemniscate, which is symmetrically placed in 1 3 quadrant like two leaves.

According to the range of θ angle 0≤θ≤∏/2, we can know that this question should take the leaves in 1 quadrant.

The area of this double integral should not be a semicircle in the figure.

This is because, if it is such a semicircle, then the polar coordinate equation of the semicircle should be r=cosθ.

Then D={(r, θ)|0≤θ≤∏/2, 0≤r≤cosθ}

Then the integral limit of iterative integral r should be from 0 to cosθ, not from 0 to √sin2θ given in the question.

Bernoulli lemniscate can be found in the first volume of Tongji Higher Mathematics.