Q:

Finding the area enclosed by an implicit function

How can I solve area of following plot by using double integral?

A:

You can convert to polar coordinates:
r[t_] := 2 Cos[t]^2
Integrate[r[t]^2/2, {t, 0, 2 Pi}]

yields $3\pi/2\approx 4.71239 $
or you can use Green's Theorem (with $\vec{F}=\{-y/2,x/2\}$):=
Integrate[{-r[t] Sin[t], r[t] Cos[t]} .D[{r[t] Cos[t], r[t] Sin[t]}, t]],{t,0,2Pi}]/2

also yielding $3\pi/2$
or approximate using ImplicitRegion:
reg = ImplicitRegion[(x^2 + y^2)^3 <= 4 x^4, {{x, -2, 2}, {y, -2, 2}}]
RegionMeasure[DiscretizeRegion[reg, MaxCellMeasure -> {"Length" -> 0.01}]]

yields: 4.71238
See Kuba comment below for shorter ImplicitRegion solution:
Area@ImplicitRegion[(x^2. + y^2)^3. <= 4. x^4., {x, y}]

yields: 4.71239

