Step by step
The question
Question: find the double integral of 1 over the region between y = x² and y = 4.
1 Draw the region
Draw the grid.
Draw y = 4
and y = x².
The region is between them.
Where do they meet?
Where the two heights are equal.
Bring everything to one side.
Move the number and divide: x² = 4.
So they meet at x = −2 and x = 2.
Which is on top?
Try x = 0: y = 4 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 4.
Outside, x goes from −2 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 = 4, then take away the bottom y = x².
So it is 4 − x².
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 = −2.
So it is 32/3.
5 Check
Let's check if it makes sense.
The region's area is 32/3.
On it, f goes from 1 to 1.
So the answer must be between 10.667 and 10.667.
10.667 is.
Right!
Let's look back.
First: draw the region.
Then: find the limits.
Then: the inner integral.
Then: the outer integral.
Last: check.