Step by step
The question
Question: find the area of one petal r = 2sin(4θ).
1 Draw it
A polar grid: circles around the center, and a line for each angle.
For each angle θ, go out a distance r.
It makes 8 petals.
We want one petal: this one.
2 One trip around
One petal starts and ends at the center, where r = 0.
sin of 4θ is 0 at θ = 0 and θ = π/4.
3 Write the integral
Cut the area into thin slices from the center.
Each is almost a thin triangle: one half r squared dθ, like a slice of a circle.
Add up the slices: A is one half of the integral of r squared.
Careful!
Do not forget the one half!
A slice is half of r squared dθ.
4 Square it
Square r.
5 Use the half-angle rule
The half-angle rule: sin squared is 1 minus cos of double the angle, over 2.
Use it, then add the plain numbers: 2.
6 Integrate
Integrate: a number gives the number times θ.
cos gives sin, divided by the number in front of θ.
7 Top minus bottom
Every sin part is 0 at these limits: sin of a whole number of half turns is 0.
Only the θ part is left: 2 times the length of the interval, π/4.
So the area is π over 4 square units, about 0.79.
8 Check
Let's check with a calculator: it adds up 1000 thin slices.
That gives about 0.79, the same as our answer.
Right!
Let's look back.
First: draw it.
Then: one trip around.
Then: write the integral.
Then: square it.
Then: use the half-angle rule.
Then: integrate.
Then: top minus bottom.
Last: check.