Step by step
The question
Question: find the double integral of x over the region between y = x² and y = x.
1 Draw the region
Draw the grid.
Draw y = x
and y = x².
The region is between them.
Where do they meet?
Where the two heights are equal.
Bring everything to one side.
Take out x.
So they meet at x = 0 and x = 1.
Which is on top?
Try x = 0.5: y = x is higher.
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 x² to x.
Outside, x goes from 0 to 1.
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 = x, then take away the bottom y = x².
So it is x² − x³.
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 = 1, then take away the bottom x = 0.
The bottom part is 0.
So it is 1/12.
5 Check
Let's check if it makes sense.
The region's area is 1/6.
On it, f goes from 0 to 1.
So the answer must be between 0 and 0.167.
0.083 is.
Right!
Let's look back.
First: draw the region.
Then: find the limits.
Then: the inner integral.
Then: the outer integral.
Last: check.