Step by step
The question
Question: estimate the area under y = 9 − x² from x = 0 to x = 3, with 3 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.
There are four choices.
First we solve it, the normal way.
Then we pick.
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.
Right rectangles: each height is at the right end.
So we use every cut but the first.
3 rectangles, 3 heights: 1, 2, 3.
Careful!
3 rectangles have 3 heights, not 4.
The first cut, 0, is not used.
The right way: x = 1, 2, 3.
Now the heights: put each x into f.
f of 1 is 8.
f of 2 is 5.
f of 3 is 0.
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 13.
So the area is about 13 square units.
5 Check
Let's check with the exact area: the integral from 0 to 3.
It is 18.
Our estimate is smaller.
That makes sense: under a falling curve, right rectangles stay below it.
The gaps are the missing area.
Right!
6 Pick the choice
Our answer is area ≈ 13 square units.
That is choice D.
7 Why not the others
A is 22.
Wrong: used left endpoints: 9 + 8 + 5.
B is 8.
Wrong: only the first rectangle.
C is 18.
Wrong: that is the exact area.
Let's look back.
First: the width.
Then: the heights.
Then: draw the rectangles.
Then: add the rectangles.
Then: check.
Then: pick the choice.
Last: why not the others.