Worked examples
The question
Question: estimate the area under y = x² from x = 0 to x = 3, with 3 left rectangles.
y = x2
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, 3 minus 0, cut into 3 equal parts.
Δx = 3 − 03
So each rectangle is 1 wide.
= 1
2 The heights
The cuts are every 1: 0, 1, 2, 3.
cuts: 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.
x = 0, 1, 2
Careful!
3 rectangles have 3 heights, not 4.
The last cut, 3, is not used.
Careful: not x = 0, 1, 2, 3 ✗
Now the heights: put each x into f.
f of 0 is 0.
The others the same way.
0
1
4
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.
area ≈ 1 × (0 + 1 + 4)
The heights add up to 5.
≈ 1 × 5
So the area is about 5 square units.
≈ 5 square units
5 Check
Let's check with the exact area: the integral from 0 to 3.
It is 9.
exact: ∫ from 0 to 3 of x2 dx = 9
Our estimate is smaller.
That makes sense: under a rising curve, left rectangles stay below it.
The gaps are the missing area.
Right!
5 < 9
Let's look back.
First: the width.
Then: the heights.
Then: draw the rectangles.
Then: add the rectangles.
Last: check.
▶ Watch it solved
The question
Question: estimate the area under y = x² from x = 0 to x = 2, with 4 left rectangles.
y = x2
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, 2 minus 0, cut into 4 equal parts.
Δx = 2 − 04
So each rectangle is 0.5 wide.
= 0.5
2 The heights
The cuts are every 0.5: 0, 0.5, 1, 1.5, 2.
cuts: 0, 0.5, 1, 1.5, 2
Left rectangles: each height is at the left end.
So we use every cut but the last.
4 rectangles, 4 heights: 0, 0.5, 1, 1.5.
x = 0, 0.5, 1, 1.5
Careful!
4 rectangles have 4 heights, not 5.
The last cut, 2, is not used.
Careful: not x = 0, 0.5, 1, 1.5, 2 ✗
Now the heights: put each x into f.
f of 0 is 0.
The others the same way.
0
0.25
1
2.25
3 Draw the rectangles
Let's draw it.
The 4 rectangles: each one 0.5 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 ≈ 0.5 × (0 + 0.25 + 1 + 2.25)
The heights add up to 3.5.
≈ 0.5 × 3.5
So the area is about 1.75 square units.
≈ 1.75 square units
5 Check
Let's check with the exact area: the integral from 0 to 2.
It is about 2.67.
exact: ∫ from 0 to 2 of x2 dx = 83 ≈ 2.67
Our estimate is smaller.
That makes sense: under a rising curve, left rectangles stay below it.
The gaps are the missing area.
Right!
1.75 < 2.67
Let's look back.
First: the width.
Then: the heights.
Then: draw the rectangles.
Then: add the rectangles.
Last: check.
Answer ≈ 1.75 square units
▶ Watch it solved
The question
Question: estimate the area under y = x² − 2x + 2 from x = 0 to x = 3, with 6 left rectangles.
y = x2 − 2x + 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, 3 minus 0, cut into 6 equal parts.
Δx = 3 − 06
So each rectangle is 0.5 wide.
= 0.5
2 The heights
The cuts are every 0.5: 0, 0.5, 1, 1.5, 2, 2.5, 3.
cuts: 0, 0.5, 1, 1.5, 2, 2.5, 3
Left rectangles: each height is at the left end.
So we use every cut but the last.
6 rectangles, 6 heights: 0, 0.5, 1, 1.5, 2, 2.5.
x = 0, 0.5, 1, 1.5, 2, 2.5
Careful!
6 rectangles have 6 heights, not 7.
The last cut, 3, is not used.
Careful: not x = 0, 0.5, 1, 1.5, 2, 2.5, 3 ✗
Now the heights: put each x into f.
f of 0 is 2.
The others the same way.
2
1.25
1
1.25
2
3.25
3 Draw the rectangles
Let's draw it.
The 6 rectangles: each one 0.5 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 ≈ 0.5 × (2 + 1.25 + 1 + 1.25 + 2 + 3.25)
The heights add up to 10.75.
≈ 0.5 × 10.75
So the area is about 5.375 square units.
≈ 5.375 square units
5 Check
Let's check with the exact area: the integral from 0 to 3.
It is 6.
exact: ∫ from 0 to 3 of (x2 − 2x + 2) dx = 6
Close to the exact area.
More, thinner rectangles get even closer.
Right!
5.375 < 6
Let's look back.
First: the width.
Then: the heights.
Then: draw the rectangles.
Then: add the rectangles.
Last: check.
Answer ≈ 5.375 square units
▶ Watch it solved
