Area between y = x^2 - 5x + 4 and the x-axis from x = 1 to x = 3 Step by Step

Step by step

The question

Question: find the area between y = x² − 5x + 4 and the x-axis from x = 1 to x = 3.

y = x² − 5x + 4

1 Draw the region

Draw the curve first.

From 1 to 3 it is below the axis: shaded red.

An integral counts area below the x-axis as negative.

But an area is never negative.

So we work out each piece on its own.

∫ below the x-axis → a minus number

2 The antiderivative

The antiderivative: add 1 to each power, divide by the new power.

F(x) = x³3 − 5 × x²2 + 4x

Now we have F(x) = x³ over 3 − 5x² over 2 + 4x.

= x³3 − 5x²2 + 4x

3 Each piece

The integral: from 1 to 3.

Top minus bottom.

∫ from 1 to 3 of f(x) dx = F(3) − F(1)

= (3³3 − 5(3)²2 + 4(3)) − (1³3 − 5(1)²2 + 4(1))

= −32 − 116

That is −10/3.

= −103

Careful!

An area is never negative.

The minus only says: below the axis.

Careful: not area = −103 ✗

4 Add the areas

−10/3 is below the axis.

Its area is the size, without the minus: 10/3.

below: area = 103

So the area is 10/3 square units.

area = 103 square units

5 Check

Let's check F: its derivative is x² − 5x + 4, the function we started with.

Right!

F′(x) = x² − 5x + 4

Let's look back.

First: draw the region.

Then: the antiderivative.

Then: each piece.

Then: add the areas.

Last: check.

Answer