SME
Grade 12 Math · Problem type
The area under a line
Use this for “Estimate the area under y = 2x + 1 from 0 to 4 with 8 rectangles, then find it exactly.”
Chart
Shaded: 0 ≤ x ≤ 4
8 right-edge rectangles of width 0.5 from x = 0 to 4: their areas add to S = 22
This is an example. Type your own values to start a new problem.
m
Slope
example
k
y-intercept
example
b
Right end
example
n
Number of rectangles
example
w
Width of each rectangle
worked out (not typed)
S
Sum of the rectangles
worked out (not typed)
A
Exact area
worked out (not typed)
Formulas
w
=
b
n
0.5
=
4
8
S
=
m
×
w
2
×
n
×
n
+ 1
2
+
k
×
b
22
=
2
×
0.5
2
×
8
×
8 + 1
2
+
1
×
4
A
=
m
×
b
2
2
+
k
×
b
20
=
2
×
4
2
2
+
1
×
4
Assumptions
•
Right-edge rectangles of width w = b ÷ n: heights m(iw) + k, for i = 1 to n.
•
Σi = n(n + 1)/2 adds them: S = mw² × n(n + 1)/2 + kb.
•
As n → ∞, S → mb²/2 + kb, the trapezoid’s area; the line stays on or above the x-axis here.
Step-by-step
WE KNOW
m = 2, k = 1, b = 4, n = 8
FIND
width of each rectangle (w), sum of the rectangles (S), exact area (A)
Step 1 · Find width of each rectangle (w)
Use w = b ÷ n
Cut
0
to
b
into
n
strips
of
equal
width.
w
=
4
8
w
=
0.5
Step 2 · Find sum of the rectangles (S)
Use S = m × w² × n × (n + 1) ÷ 2 + k × b
Each
rectangle
is
w
by
m
(iw)
+
k
;
the
m
parts
add
to
mw²(1
+
2
+
…
+
n
),
the
k
parts
to
kb.
S
=
2
×
0.5
2
×
8
×
8 + 1
2
+
1
×
4
1
+
2
+
…
+
8
=
8
×
9
2
=
36
S
=
2
×
0.5
2
×
36
+
1
×
4
S
=
18
+
4
S
=
22
Step 3 · Find exact area (A)
Use A = m × b² ÷ 2 + k × b
The
limit:
n
(
n
+ 1)
n
2
goes
to
1.
It
is
also
the
trapezoid
with
heights
k
and
mb
+
k
.
A
=
2
×
4
2
2
+
1
×
4
A
=
2
×
16
2
+
4
A
=
32
2
+
4
A
=
16
+
4
A
=
20
→
S
=
22
is
above
the
exact
area;
more
rectangles
bring
S
closer
to
it
Check
Put the numbers back in. Both sides should match.
0.5
=
4
8
✓
22
=
2
×
0.5
2
×
8
×
8 + 1
2
+
1
×
4
✓
20
=
2
×
4
2
2
+
1
×
4
✓
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Limits at infinity by degree
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