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.