Question

How do you find the area of the region bounded by the polar curve `r=2+cos(2theta)` ?

Answer 1

The area inside a polar curve is approximately the sum of lots of skinny wedges that start at the origin and go out to the curve, as long as there are no self-intersections for your polar curve.

Each wedge or slice or sector is like a triangle with height `r` and base `r` `dθ`, so the area of each element is
`dA = 1/2 b h = 1/2 r (r dθ) = 1/2 r^2 dθ.`

So add them up as an integral going around from θ=0 to θ=2π, and using a double angle formula, we get:

`A = 1/2 int_0 ^(2π)(2 + cos(2θ))^2 dθ`

`A = 1/2 int_0 ^(2π) [4 + 4 cos(2θ) + cos^2(2θ)] dθ`

`A = 1/2 int_0 ^(2π) [4 + 4 cos(2θ) + (1 + cos(4θ))/2] dθ.`

Now do the integral(s) by subbing u = 2θ and then u = 4θ, and remember to change limits for the "new u." I'll let you evaluate those to get practice integrating! Remember our motto,

"Struggling a bit makes you stronger." \dansmath/