Type her answer in the space below and give 3 decimal places. If her answer is less than 1, ar a top "0" prior to the decimal suggest (ex: 0.482)

r = 0 when cos(θ) = -1/2, which occurs at θ = 2π/3 and 4π/3

We use I for the integral native a come b and E for the testimonial from a to b

Then, as r dr dθ = dydx, us integrate

I<2π/3, 4π/3>I<0,2+4 cos(θ)> r dr dθ =

I<2π/3, 4π/3> 1/2r2E<0,2+4 cos(θ)> dθ =

I<2π/3, 4π/3> 2 + 8 cos(θ) + 8 cos2(θ) dθ

As cos(2θ) = 2 cos2(θ) - 1, 8 cos2(θ) = 4 cos(2θ) + 4

Then, I<2π/3, 4π/3> 2 + 8 cos(θ) + 8 cos2(θ) dθ =

I<2π/3, 4π/3> 6 + 8 cos(θ) + 4 cos(2θ) dθ =

6θ + 8 sin(θ) + 2 sin(2θ) E<2π/3, 4π/3> =

4π - 8√3 + 2√3 =

4π - 6√3 =

2.174

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