Step by step
The question
Question: estimate the area under y = 4x − x² from x = 0 to x = 4, with 4 midpoint 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, 4 minus 0, cut into 4 equal parts.
So each rectangle is 1 wide.
2 The heights
The cuts are every 1: 0, 1, 2, 3, 4.
Midpoint rectangles: each height is in the middle, halfway between two cuts.
4 rectangles, 4 heights: 0.5, 1.5, 2.5, 3.5.
Careful!
Those are the left ends.
Midpoint means halfway between the cuts.
Now the heights: put each x into f.
f of 0.5 is 1.75.
The others the same way.
3 Draw the rectangles
Let's draw it.
The 4 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 11.
So the area is about 11 square units.
5 Check
Let's check with the exact area: the integral from 0 to 4.
It is about 10.67.
Close to the exact area.
More, thinner rectangles get even closer.
Right!
Let's look back.
First: the width.
Then: the heights.
Then: draw the rectangles.
Then: add the rectangles.
Last: check.