Estimate the area under y = x + 2 from x = 0 to x = 4 with 2… Step by Step

Step by step

The question

Question: estimate the area under y = x + 2 from x = 0 to x = 4, with 2 midpoint rectangles.

y = x + 2

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, 4 minus 0, cut into 2 equal parts.

Δx = 4 − 02

So each rectangle is 2 wide.

= 2

2 The heights

The cuts are every 2: 0, 2, 4.

cuts: 0, 2, 4

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

2 rectangles, 2 heights: 1, 3.

x = 1, 3

Careful!

Those are the left ends.

Midpoint means halfway between the cuts.

Careful: not x = 0, 2 ✗

Now the heights: put each x into f.

f of 1 is 3.

The others the same way.

3 Draw the rectangles

Let's draw it.

The 2 rectangles: each one 2 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 ≈ 2 × (3 + 5)

The heights add up to 8.

≈ 2 × 8

So the area is about 16 square units.

≈ 16 square units

5 Check

Let's check with the exact area: the integral from 0 to 4.

It is 16.

exact: ∫ from 0 to 4 of (x + 2) dx = 16

Close to the exact area.

More, thinner rectangles get even closer.

Right!

16 = 16

Let's look back.

First: the width.

Then: the heights.

Then: draw the rectangles.

Then: add the rectangles.

Last: check.

Answer