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AP Calculus BC
Unit 6: Integration and Accumulation of Change
6.2 Approximating Areas with Riemann Sums
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What is a Riemann Sum used for?
Approximating the area under a curve
A Riemann Sum divides the interval
[
a
,
b
]
[a, b]
[
a
,
b
]
into
n
n
n
subintervals of equal width
There are three types of Riemann Sums: Left, Right, and
Midpoint
How is the height determined in a Left Riemann Sum?
Left endpoint of each subinterval
How is the height determined in a Right Riemann Sum?
Right endpoint of each subinterval
How is the height determined in a Midpoint Riemann Sum?
Midpoint of each subinterval
Using
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
x^{2}
x
2
on the interval
[
0
,
2
]
[0, 2]
[
0
,
2
]
with
n
=
n =
n
=
4
4
4
, the width
Δ
x
\Delta x
Δ
x
is 0.5
The Left Riemann Sum for
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
x^{2}
x
2
on
[
0
,
2
]
[0, 2]
[
0
,
2
]
with
n
=
n =
n
=
4
4
4
is 1.25
Which endpoint is used to find the height in a Left Riemann Sum?
Left endpoint
Which endpoint is used to find the height in a Right Riemann Sum?
Right endpoint
Which point is used to find the height in a Midpoint Riemann Sum?
Midpoint
The formula for a Left Riemann Sum is
Δ
x
∑
i
=
1
n
f
(
x
i
−
1
)
\Delta x \sum_{i = 1}^{n} f(x_{i - 1})
Δ
x
∑
i
=
1
n
f
(
x
i
−
1
)
, where
Δ
x
\Delta x
Δ
x
is the width of each subinterval.
The Right Riemann Sum formula uses the right endpoint
x
i
x_{i}
x
i
to calculate the height
What is the formula for calculating a Left Riemann Sum?
\sum_{i = 1}^{n} f(x_{i - 1}) \Delta x</latex>
In the Left Riemann Sum formula,
Δ
x
\Delta x
Δ
x
is calculated as
b
−
a
n
\frac{b - a}{n}
n
b
−
a
, where
a
a
a
and
b
b
b
are the limits of integration.
What does
n
n
n
represent in the Left Riemann Sum formula?
Number of subintervals
Steps to calculate a Left Riemann Sum for
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
x^{2}
x
2
on [0, 2]</latex> with
n
=
n =
n
=
4
4
4
1️⃣ Calculate
Δ
x
=
\Delta x =
Δ
x
=
2
−
0
4
=
\frac{2 - 0}{4} =
4
2
−
0
=
0.5
0.5
0.5
2️⃣ Determine the left endpoints: 0, 0.5, 1, 1.5
3️⃣ Calculate
f
(
x
i
−
1
)
f(x_{i - 1})
f
(
x
i
−
1
)
for each left endpoint
4️⃣ Multiply each
f
(
x
i
−
1
)
f(x_{i - 1})
f
(
x
i
−
1
)
by
Δ
x
\Delta x
Δ
x
5️⃣ Sum the results to approximate the area
What is the approximate area under
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
x^{2}
x
2
on
[
0
,
2
]
[0, 2]
[
0
,
2
]
using a Left Riemann Sum with
n
=
n =
n
=
4
4
4
?
1.75
The Left Riemann Sum uses the left
endpoints
The formula for the Left Riemann Sum is \sum_{i =1}^{n} f(x_{i - 1})
\Delta x
</latex>
What does
Δ
x
\Delta x
Δ
x
represent in the Left Riemann Sum formula?
Width of each subinterval
In the Left Riemann Sum,
x
i
−
1
x_{i - 1}
x
i
−
1
is the left endpoint
Steps to approximate the area under
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
x^{2}
x
2
on
[
0
,
2
]
[0, 2]
[
0
,
2
]
using the Left Riemann Sum with
n
=
n =
n
=
4
4
4
1️⃣ Calculate
Δ
x
\Delta x
Δ
x
2️⃣ Determine the left endpoints
3️⃣ Evaluate
f
(
x
i
−
1
)
f(x_{i - 1})
f
(
x
i
−
1
)
4️⃣ Calculate
f
(
x
i
−
1
)
Δ
x
f(x_{i - 1}) \Delta x
f
(
x
i
−
1
)
Δ
x
5️⃣ Sum the areas to find the total
What is the value of
Δ
x
\Delta x
Δ
x
when approximating the area under
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
x^{2}
x
2
on
[
0
,
2
]
[0, 2]
[
0
,
2
]
with
n
=
n =
n
=
4
4
4
?
0.5
The approximate area under f(x) = x^{2}</latex> on
[
0
,
2
]
[0, 2]
[
0
,
2
]
using the Left Riemann Sum with
n
=
n =
n
=
4
4
4
is 1.75.
A Riemann Sum approximates the area under a curve by dividing the interval into
subintervals
What are the three types of Riemann Sums?
Left, Right, Midpoint
Match the Riemann Sum type with its defining characteristic:
Left Riemann Sum ↔️ Uses left endpoint
Right Riemann Sum ↔️ Uses right endpoint
Midpoint Riemann Sum ↔️ Uses midpoint of subinterval
The Right Riemann Sum uses the right
endpoint
of each subinterval to calculate the height of the rectangles.
The Midpoint Riemann Sum uses the midpoint of each
subinterval
Match the Riemann Sum type with its height calculation:
Left Riemann Sum ↔️ f(x_{i - 1})</latex>
Right Riemann Sum ↔️
f
(
x
i
)
f(x_{i})
f
(
x
i
)
Midpoint Riemann Sum ↔️
f\left(\frac{x_{i - 1} +
x_{i}}{2}\right)
In the Left Riemann Sum, the height of each rectangle is calculated using
f
(
x
i
−
1
)
f(x_{i - 1})
f
(
x
i
−
1
)
In the Midpoint Riemann Sum, the height is calculated using the midpoint formula
f\left(\frac{x_{i - 1} +
x_{i}}{2}\right)
True
What is a Riemann Sum used to approximate?
Area under a curve
There are three types of Riemann Sums: Left, Right, and
Midpoint
Match the Riemann Sum type with its endpoint used:
Left Riemann Sum ↔️ Left endpoint
Right Riemann Sum ↔️ Right endpoint
Midpoint Riemann Sum ↔️ Midpoint
What is the height calculation for a Left Riemann Sum at the
i
i
i
-th subinterval?
f(x_{i - 1})</latex>
What is the height calculation for a Right Riemann Sum at the
i
i
i
-th subinterval?
f
(
x
i
)
f(x_{i})
f
(
x
i
)
The Midpoint Riemann Sum uses the midpoint of each
subinterval
to calculate the height.
The Midpoint Riemann Sum provides a more accurate approximation than the
Left or Right Riemann Sums
.
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