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AP Calculus BC
Unit 6: Integration and Accumulation of Change
6.1 Exploring Accumulations of Change
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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
Trapezoidal Rule ↔️ Uses trapezoids
The formula for the Left Riemann Sum is \sum_{i = 1}^{n} f(x_{i - 1}) \Delta
x
The Right Riemann Sum uses the right
endpoint
of each subinterval to determine the height of the rectangle.
The formula for the Midpoint Riemann Sum is \sum_{i = 1}^{n} f\left(\frac{x_{i - 1} + x_{i}}{2}\right) \Delta
x
What shape is used to approximate the area under a curve in the Trapezoidal Rule?
Trapezoids
Approximating
∫
0
2
x
2
d
x
\int_{0}^{2} x^{2} dx
∫
0
2
x
2
d
x
using the Left Riemann Sum with 4 subintervals gives 2.75.
Order the Riemann sum types from lowest to highest approximation of
∫
0
2
x
2
d
x
\int_{0}^{2} x^{2} dx
∫
0
2
x
2
d
x
using 4 subintervals.
1️⃣ Left Riemann Sum
2️⃣ Midpoint Riemann Sum
3️⃣ Trapezoidal Rule
4️⃣ Right Riemann Sum
What mathematical concept is used to represent the accumulation of change?
Definite integral
The integral
∫
a
b
f
(
x
)
d
x
\int_{a}^{b} f(x) dx
∫
a
b
f
(
x
)
d
x
calculates the accumulated change of a function
f
(
x
)
f(x)
f
(
x
)
from
x
=
x =
x
=
a
a
a
to x = b
In the integral
∫
a
b
f
(
x
)
d
x
\int_{a}^{b} f(x) dx
∫
a
b
f
(
x
)
d
x
,
f
(
x
)
f(x)
f
(
x
)
represents the rate of change.
The definite integral
∫
0
5
3
t
2
d
t
\int_{0}^{5} 3t^{2} dt
∫
0
5
3
t
2
d
t
equals 125
What are Riemann sums used to approximate?
Definite integrals
Riemann sums divide the interval into subintervals and sum the areas of rectangles or
trapezoids
.
What does the symbol > represent in calculus?
Rate of change
To find the total water accumulated from
t
=
t =
t
=
0
0
0
to
t
=
t =
t
=
5
5
5
minutes for
f
(
t
)
=
f(t) =
f
(
t
)
=
3
t
2
3t^{2}
3
t
2
, we calculate the definite integral.
125
liters
of water accumulated in the tank.
What are Riemann Sums used to approximate?
Definite integrals
The Left Riemann Sum uses the left endpoint of each subinterval to determine the height of the
rectangle
.
Match the Riemann Sum type with its characteristic:
Left Riemann Sum ↔️ Uses left endpoint
Right Riemann Sum ↔️ Uses right endpoint
Midpoint Riemann Sum ↔️ Uses midpoint
Trapezoidal Rule ↔️ Uses trapezoids
The Trapezoidal Rule approximates the area using
trapezoids
instead of rectangles.
Approximate
∫
0
2
x
2
d
x
\int_{0}^{2} x^{2} dx
∫
0
2
x
2
d
x
using 4 subintervals and Riemann Sums in increasing order of approximation:
1️⃣ Left Riemann Sum: 2.75
2️⃣ Midpoint Riemann Sum: 3.375
3️⃣ Trapezoidal Rule: 3.75
4️⃣ Right Riemann Sum: 4.75
Which Riemann Sum type gives an overestimate for
∫
0
2
x
2
d
x
\int_{0}^{2} x^{2} dx
∫
0
2
x
2
d
x
using 4 subintervals?
Right Riemann Sum
The Fundamental Theorem of Calculus Part 1 states that if
F
(
x
)
=
F(x) =
F
(
x
)
=
∫
a
x
f
(
t
)
d
t
\int_{a}^{x} f(t) dt
∫
a
x
f
(
t
)
d
t
, thenF'(x) = f(x)</latex> is the derivative of F(x).
What does the Fundamental Theorem of Calculus Part 2 allow us to calculate?
Definite integrals
To find the area under
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
x^{2}
x
2
from
0
0
0
to
2
2
2
, we calculate
∫
0
2
x
2
d
x
=
\int_{0}^{2} x^{2} dx =
∫
0
2
x
2
d
x
=
8
3
\frac{8}{3}
3
8
, which uses the Fundamental Theorem of Calculus Part 2.
If F(x) = \int_{0}^{x} t^{3} dt</latex>, then
F
′
(
x
)
=
F'(x) =
F
′
(
x
)
=
x
3
x^{3}
x
3
.
What is the formula for finding the volume of a solid using cross-sectional areas?
V
=
V =
V
=
∫
a
b
A
(
x
)
d
x
\int_{a}^{b} A(x) dx
∫
a
b
A
(
x
)
d
x
What does the Fundamental Theorem of Calculus (FTC) connect?
Differentiation and integration
FTC Part 1 states that if F(x) = \int_{a}^{x} f(t) dt</latex>, then <latex>F'(x) = f(x)
FTC Part 2 calculates the definite integral by finding an antiderivative
F
(
x
)
F(x)
F
(
x
)
such that
F
′
(
x
)
=
F'(x) =
F
′
(
x
)
=
f
(
x
)
f(x)
f
(
x
)
and evaluating
F
(
b
)
−
F
(
a
)
F(b) - F(a)
F
(
b
)
−
F
(
a
)
.
To find the area under
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
x^{2}
x
2
from
0
0
0
to
2
2
2
, we calculate \int_{0}^{2} x^{2} dx = \frac{8}{3}
Match the concept with its formula:
FTC Part 1 ↔️
F
′
(
x
)
=
F'(x) =
F
′
(
x
)
=
f
(
x
)
f(x)
f
(
x
)
FTC Part 2 ↔️
∫
a
b
f
(
x
)
d
x
=
\int_{a}^{b} f(x) dx =
∫
a
b
f
(
x
)
d
x
=
F
(
b
)
−
F
(
a
)
F(b) - F(a)
F
(
b
)
−
F
(
a
)
Area ↔️
A
=
A =
A
=
∫
a
b
f
(
x
)
d
x
\int_{a}^{b} f(x) dx
∫
a
b
f
(
x
)
d
x
Volume ↔️
V
=
V =
V
=
∫
a
b
A
(
x
)
d
x
\int_{a}^{b} A(x) dx
∫
a
b
A
(
x
)
d
x
Steps to calculate accumulated change using a definite integral:
1️⃣ Identify the rate of change function
f
(
x
)
f(x)
f
(
x
)
2️⃣ Determine the interval
[
a
,
b
]
[a, b]
[
a
,
b
]
3️⃣ Calculate the definite integral
∫
a
b
f
(
x
)
d
x
\int_{a}^{b} f(x) dx
∫
a
b
f
(
x
)
d
x
4️⃣ Interpret the result as the total change
Riemann sums approximate the definite integral by dividing the interval into subintervals and summing the areas of rectangles or
trapezoids
.
In a left Riemann sum, the height of each rectangle is determined by the
left
endpoint of the subinterval.
Match the Riemann sum type with its formula:
Left Riemann Sum ↔️
∑
i
=
1
n
f
(
x
i
−
1
)
Δ
x
\sum_{i = 1}^{n} f(x_{i - 1}) \Delta x
∑
i
=
1
n
f
(
x
i
−
1
)
Δ
x
Right Riemann Sum ↔️
∑
i
=
1
n
f
(
x
i
)
Δ
x
\sum_{i = 1}^{n} f(x_{i}) \Delta x
∑
i
=
1
n
f
(
x
i
)
Δ
x
What is the formula for the Right Riemann Sum?
∑
i
=
1
n
f
(
x
i
)
Δ
x
\sum_{i = 1}^{n} f(x_{i}) \Delta x
∑
i
=
1
n
f
(
x
i
)
Δ
x
The Right Riemann Sum uses the
right
endpoint of each subinterval to determine the height of the rectangle.
Riemann Sums are used to approximate the value of a
definite integral
.
What is the formula for the Left Riemann Sum?
∑
i
=
1
n
f
(
x
i
−
1
)
Δ
x
\sum_{i = 1}^{n} f(x_{i - 1}) \Delta x
∑
i
=
1
n
f
(
x
i
−
1
)
Δ
x
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