Find the area of one petal on the given graph. Round answer to two decimal places.

Question

Find the area of one petal on the given graph. Round answer to two decimal places.

The area of one petal is 

Answer 1

r = 5sin(3θ)

When θ = 0, r = 0

When θ = π/6, r = 5

When θ = π/3, r = 0

You can see the graph on Wolfram Alpha, for example.

The graph has 3 loops, each of which has the same area.

One loop is traced out as θ varies from 0 to π/3.

Area of loop = (1/2)∫ r²dθ = (25/2)∫_{(0 to π/3)} sin²(3θ)dθ = (25/4)∫_{(0 to π/3)} (1 - cos(6θ))dθ

= (25/4) [ θ - (1/6)sin(6θ) ]_{(0 to π/3)} = (25/4) [ (π/3 - 0) - (0) ] = 25π/12