Step by step
The question
Question: estimate the area under y = x² from x = 0 to x = 3, with 3 left 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, 3 minus 0, cut into 3 equal parts.
So each rectangle is 1 wide.
2 The heights
The cuts are every 1: 0, 1, 2, 3.
Left rectangles: each height is at the left end.
So we use every cut but the last.
3 rectangles, 3 heights: 0, 1, 2.
Careful!
3 rectangles have 3 heights, not 4.
The last cut, 3, is not used.
Now the heights: put each x into f.
f of 0 is 0.
The others the same way.
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.
The heights add up to 5.
So the area is about 5 square units.
5 Check
Let's check with the exact area: the integral from 0 to 3.
It is 9.
Our estimate is smaller.
That makes sense: under a rising curve, left rectangles stay below it.
The gaps are the missing area.
Right!
Let's look back.
First: the width.
Then: the heights.
Then: draw the rectangles.
Then: add the rectangles.
Last: check.