Step by step
The question
Question: estimate the area under y = x² + 1 from x = 0 to x = 2, with 2 right rectangles.
A rectangle's area is easy: width times height.
So we fill the area under the curve with rectangles, and add them up.
1 The width
Each rectangle has the same width.
We call it Δx, "delta x".
The whole width, 2 minus 0, cut into 2 equal parts.
So each rectangle is 1 wide.
2 The heights
The cuts are every 1: 0, 1, 2.
Right rectangles: each height is at the right end.
So we use every cut but the first.
2 rectangles, 2 heights: 1, 2.
Careful!
2 rectangles have 2 heights, not 3.
The first cut, 0, is not used.
Now the heights: put each x into f.
f of 1 is 2.
The others the same way.
3 Draw the rectangles
Let's draw it.
The 2 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.
The heights add up to 7.
So the area is about 7 square units.
5 Check
Let's check with the exact area: the integral from 0 to 2.
It is about 4.67.
Our estimate is bigger.
That makes sense: on a rising curve, right rectangles stick out above it.
Right!
Let's look back.
First: the width.
Then: the heights.
Then: draw the rectangles.
Then: add the rectangles.
Last: check.