Step by step
The question
Question: find the area of one petal r = cos(3θ).
The area inside a curve written with r and θ.
There are four choices.
First we solve it, the normal way.
Then we pick.
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 3 petals.
We want one petal: this one.
2 One trip around
One petal starts and ends at the center, where r = 0.
cos of 3θ is 0 when 3θ is minus π/2 or π/2.
Divide by 3: θ = −π/6 and θ = π/6.
3 Write the integral
Cut the area into thin slices from the center.
Each is almost a thin triangle: one half r squared dθ.
A slice of a circle with angle θ has area one half r squared θ.
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.
The right way: A = 1/2∫ from −π/6 to π/6 of (cos(3θ))² dθ.
4 Square it
Square r.
5 Use the half-angle rule
The half-angle rule: cos squared is 1 plus cos of double the angle, over 2.
Use it, then add the plain numbers: 1/2.
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: 1/2 times the length of the interval, π/3.
So the area is π over 12 square units, about 0.26.
8 Check
Let's check with a calculator: it adds up 1000 thin slices.
That gives about 0.26, the same as our answer.
Right!
9 Pick the choice
Our answer is A = π over 12.
That is choice B.
10 Why not the others
A is π/6.
Wrong: lost the 1/2 of the area formula.
That is the Careful mistake from before.
C is π/3.
Wrong: that is the angle of one petal.
D is π/24.
Wrong: halved twice.
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.
Then: check.
Then: pick the choice.
Last: why not the others.