Find the area of the region cut from the first quadrant by the curve

Question

r=2(2-sin2ɵ)½

Answer 1

I suspect you mean:
r=2(2-sin2?)^½
that makes me nervous that since you do not type exponents you might mean 2-sin^2 ?
instead of what your typed. I will assume that

first graph it

if the latter
da = (1/2)r^2 d?
where
r^2=4(2-sin^2?)

da = 2 (2-sin^2 ?) d?

= 4 d? - 2 sin^2 ? d? zero to pi/2

= 4 ? - 2 [?/2 - sin2? /4 ]

=4(pi/2) -2 [pi/4 - 0]

= 2 pi - pi/2 = 3 pi/2
check my arithmetic and if I understood what you typed