Area between y = 1 and y = x^2

Step by step

The question

Question: find the area between y = 1 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.

x² − 1 is a difference of two squares: x² minus 1 squared.

So it is (x − 1)(x + 1).

So x = −1 or x = 1.

3 The top curve

Test a number between −1 and 1: x = 0.

The first gives 1, the second 0.

So y = 1 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.

6 Top minus bottom

Put in 1, then take away the same with −1.

So the area is 4/3 square units.

7 Check

Let's check F: its derivative is −x² + 1: 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.