Estimate the area under y = x^2 from x = 0 to x = 3 with 3 midpoint… Step by Step

Step by step

The question

Question: estimate the area under y = x² from x = 0 to x = 3, with 3 midpoint rectangles.

y = x²

A rectangle's area is easy: width times height.

So we fill the area under the curve with rectangles, and add them up.

rectangle area = width × height

1 The width

Each rectangle has the same width.

We call it Δx, "delta x".

Δx = the width of each rectangle

The whole width, 3 minus 0, cut into 3 equal parts.

Δx = 3 − 03

So each rectangle is 1 wide.

= 1

2 The heights

The cuts are every 1: 0, 1, 2, 3.

cuts: 0, 1, 2, 3

Midpoint rectangles: each height is in the middle, halfway between two cuts.

3 rectangles, 3 heights: 0.5, 1.5, 2.5.

x = 0.5, 1.5, 2.5

Careful!

Those are the left ends.

Midpoint means halfway between the cuts.

Careful: not x = 0, 1, 2 ✗

Now the heights: put each x into f.

f of 0.5 is 0.25.

The others the same way.

0.25

2.25

6.25

3 Draw the rectangles

Let's draw it.

The 3 rectangles: each one 1 wide, with its height at the green dot.

4 Add the rectangles

Every rectangle has the same width, so add the heights, then times the width.

area ≈ 1 × (0.25 + 2.25 + 6.25)

The heights add up to 8.75.

≈ 1 × 8.75

So the area is about 8.75 square units.

≈ 8.75 square units

5 Check

Let's check with the exact area: the integral from 0 to 3.

It is 9.

exact: ∫ from 0 to 3 of x² dx = 9

Close to the exact area.

More, thinner rectangles get even closer.

Right!

8.75 < 9

Let's look back.

First: the width.

Then: the heights.

Then: draw the rectangles.

Then: add the rectangles.

Last: check.

Answer