Find the area enclosed by the graph of r = 4 cos 3θ.Type your answer in the space below and give 3 decimal places. If your answer is less than 1, place a leading "0" before the decimal point

Question

Find the area enclosed by the graph of r = 4 cos 3θ.Type your answer in the space below and give 3 decimal places. If your answer is less than 1, place a leading "0" before the decimal point (ex: 0.482)

Answer 1

You should graph the 3-petalled rose curve in the plane to establish the bounds of integration. Although the period of cos3Θ is 2π/3, we can see from the graph that if we let Θ range from 0 to π the whole graph is drawn. So we can set those as the bounds of integration.

For a polar area, we use the formula A =1/2 ∫_a^b r²dΘ:

A = 1/2 ∫₀^π (4cos(3θ))²dΘ = 8 ∫₀^π (cos²(3θ))dΘ . Now we need a double-angle trig substitution to make the function integrable: cos²(3θ) = 1/2(cos(6Θ) + 1)

A = 4 ∫₀^π (cos6Θ + 1)dΘ = 4(1/6sin6Θ + Θ)]₀^π = 4π