Question

find the area of enclosed by the gievn curve.

r= a cos 2(θ)

Answer 1

Look at this similar example have just solved:
Find the area enclosed by the curve:

r=2+3cosθ
Solution:
Area of curve,
A=2∫arccos(−2/3)0r22dϕ
⟹A=2∫arccos(−2/3)0(2+3cosϕ)22dϕ
=2∫arccos(−2/3)04+9cos2ϕ+12cosϕ2dϕ
=∫arccos(−2/3)0(4+9cos2ϕ+12cosϕ)dϕ
=35√+172cos−1(−23).
Does it make some sense to you?

Answer 2

There is a symmetry about x-axis.Do u think so?

Answer 3

If \(r=a\cos^2\theta\), then the area would be
\[A=\frac{1}{2}\int_a^br^2~d\theta=\frac{1}{2}\int_0^{2\pi}\left(2\cos^2\theta\right)^2~d\theta\\
A=2\int_0^{2\pi}\cos^4\theta~d\theta\]
Use the half-angle identity:
\[\cos^{2}x=\frac{1}{2}\left(1+\cos2x\right)\]
So you have
\[\cos^{4}x=\left(\cos^2x\right)^2=\left(\frac{1}{2}\left(1+\cos2x\right)\right)^2\]