Step by step
The question
Question: find the area inside the curve r = 2 + sin θ.
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
The curve goes around once as θ goes 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 the bracket: first squared, two times first times second, second squared.
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: 4 plus 1/2 is 9/2.
6 Integrate
Integrate: a number gives the number times θ.
cos gives sin, divided by the number in front of θ; sin gives minus cos.
7 Top minus bottom
Every sin part is 0 at these limits: sin of a whole number of half turns is 0.
And cos 2π − cos 0 is 0.
Only the θ part is left: 9/2 times the length of the interval, 2π.
So the area is 9π over 2 square units, about 14.14.
8 Check
Let's check with a calculator: it adds up 1000 thin slices.
That gives about 14.14, 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.