Step by step
The question
Question: find the double integral of x² over 0 ≤ x ≤ 2, 0 ≤ y ≤ 2x.
1 Draw the region
Draw the grid.
Draw y = 2x
and y = 0.
The region is between them.
2 Find the limits
Take a thin upright strip.
On it, y goes from the bottom curve up to the top curve.
So inside, y goes from 0 to 2x.
Outside, x goes from 0 to 2.
Careful!
The inside integral is dy, so its limits are the y limits.
They may have x in them.
The outside limits are only numbers.
3 The inner integral
Inside first: integrate in y.
x stays fixed, like a plain number.
Integrate each term: the power goes up by 1, and we divide by the new power.
Put in the top y = 2x, then take away the bottom y = 0.
The bottom part is 0.
So it is 2x³.
4 The outer integral
Now the outside integral, in x.
Its limits are numbers.
Integrate each term: the power goes up by 1, and we divide by the new power.
Put in the top x = 2, then take away the bottom x = 0.
The bottom part is 0.
So it is 8.
5 Check
Let's check if it makes sense.
The region's area is 4.
On it, f goes from 0 to 4.
So the answer must be between 0 and 16.
8 is.
Right!
Let's look back.
First: draw the region.
Then: find the limits.
Then: the inner integral.
Then: the outer integral.
Last: check.