Step by step
The question
Question: find the double integral of x² + y² over the ring 1 ≤ x² + y² ≤ 4.
1 Draw the region
The region on the floor is a ring: between radius 1 and radius 2, all the way around.
Above it is the surface z = x² + y².
The integral is the volume of this solid.
2 Change to polar
In polar form, x² + y² = r², and the small area is dA = r dr dθ.
Why the extra r?
The small pieces far from the center are wider.
Their width grows with r.
r goes from 1 to 2.
θ goes from 0 to 2π: once all the way around.
Write it in r and θ.
x² + y² becomes r².
Then times r dr dθ.
Careful!
Without the extra r, the answer is wrong.
It is the most common mistake here.
3 The inner integral
Multiply by r first: r³.
Integrate each term: the power goes up by 1, and we divide by the new power.
Put in the top r = 2, then take away the bottom r = 1.
So it is 15/4.
4 The outer integral
Now θ.
15/4 is a plain number: its integral is 15/4 times θ.
Top minus bottom: 2π minus 0 is 2π.
So the answer is 15π/2.
That is about 23.56.
5 Check
Let's check if it makes sense.
The region's area is 3π.
On it, f goes from 1 to 4.
So the answer must be between 9.42 and 37.7.
23.56 is.
Right!
Let's look back.
First: draw the region.
Then: change to polar.
Then: the inner integral.
Then: the outer integral.
Last: check.