Step by step
The question
Question: find the area between y = 2x and y = x².
1 Draw the region
Draw both curves.
They cross twice.
The area we want is between the crossings.
2 Where they meet
They meet where the two y's are equal.
Move everything to one side.
Factor it.
We need two numbers that multiply to 0 and add to −2: 0 and −2.
So x = 0 or x = 2.
3 The top curve
Test a number between 0 and 2: x = 1.
The first gives 2, the second 1.
So y = 2x is on top.
The area between them is shaded.
4 Write the integral
Area between = the integral of the top one minus the bottom one.
Careful!
Top minus bottom!
The other way gives a minus answer, −4/3.
A minus in front of a bracket changes the sign of every part inside.
Then like terms.
5 The antiderivative
The antiderivative: add 1 to each power, divide by the new power.
Now we have F(x) = −x³ over 3 + x².
6 Top minus bottom
Put in 2, then take away the same with 0.
So the area is 4/3 square units.
7 Check
Let's check F: its derivative is −x² + 2x: top minus bottom.
And the area is positive.
Right!
Let's look back.
First: draw the region.
Then: where they meet.
Then: the top curve.
Then: write the integral.
Then: the antiderivative.
Then: top minus bottom.
Last: check.