Step by step
The question
Question: find the area inside the curve r = 2.
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.
The curve closes up.
2 One trip around
A circle around the center: once around is θ from 0 to 2π.
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 Integrate
Integrate: a number gives the number times θ.
cos gives sin, divided by the number in front of θ.
6 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: 4 times the length of the interval, 2π.
So the area is 4π square units, about 12.57.
7 Check
Let's check: the curve is a circle, radius 2.
π r squared gives 4π.
The same.
Right!
Let's look back.
First: draw it.
Then: one trip around.
Then: write the integral.
Then: square it.
Then: integrate.
Then: top minus bottom.
Last: check.