CSUN MATH 150A: Comprehensive Study Guide on The
Cumulative Effect of a Function
(Accumulation and the Definite Integral)
A CSUN Students Insight on Mastering Accumulation
As a Math 150A student transitioning into integrals, this topic—"The Cumulative
Effect of a Function"—is arguably the single most important conceptual leap in
calculus. Its the moment we move from instantaneous rates of change
(derivatives) to total change (integrals). Think of it this way: The derivative f ′ ( x)
tells you how fast your cars position is changing right now (velocity). The integral
b
∫ f ′ ( x) dx tells you the total distance the car traveled between time a and time b.
a
The whole idea boils down to: Area under a rate function equals total
accumulated quantity. We approximate this area using the sum of small rectangles
(Riemann Sums), and then we take the limit as the rectangles become infinitely thin
—thats the magic of the definite integral. Master the Fundamental Theorem of
Calculus (FTC), and you master accumulation.
I. Foundational Concepts: What is Accumulation?
A. The Problem of Area
Before calculus, to find the area of a shape, we needed a formula (e.g.,
A=length × width ). The concept of accumulation begins with the problem of finding
the area under a non-linear curve, f ( x), between two points, x=a and x=b .
When f (x) represents a rate of change (e.g., velocity in miles per hour), the area
under the curve represents the total change or accumulation of the original
quantity (e.g., total distance in miles).
B. Riemann Sums: The Accumulation Approximation
Since we cant find the area exactly with simple geometry, we approximate it using
rectangles. This process is called a Riemann Sum.
1. Partitioning the Interval: Divide the interval [a , b] into n subintervals of
equal width, Δ x .
, b−a
Δ x=
n
2. Choosing Sample Points: Within each subinterval, [ xi − 1 , x i] , we choose a
sample point x ∗i . The height of the i -th rectangle is determined by the
function value at this point, f ( x ∗i ).
3. Calculating the Area of Rectangles: The area of the i -th rectangle is
∗
Ai=f (x i ) ⋅ Δ x .
4. Summing the Areas: The total approximate area, Rn, is the sum of the areas
of all n rectangles:
n
Rn =∑ f ( x∗i ) ⋅ Δ x
i=1
The common choices for x ∗i are:
Left Endpoint Sum ( Ln): x ∗i =xi − 1
Right Endpoint Sum ( Rn ): x i =xi
∗
x i −1 + x i
Midpoint Sum ( M n): x ∗i =
2
C. The Definite Integral: The Limit of Accumulation
To move from the approximation (Riemann Sum) to the exact cumulative effect, we
must let the number of rectangles (n ) approach infinity, which simultaneously
forces the width of the rectangles ( Δ x ) to approach zero.
The Definite Integral is formally defined as the limit of the Riemann Sum:
b n
∫ f (x )dx= n→
lim ∑ f (x ∗i )⋅ Δ x
∞
a i =1
Notation Breakdown:
o ∫ (The Integral Sign): An elongated "S" representing "Sum" (the limit
of the summation).
o a and b (Limits of Integration): Define the interval over which
accumulation occurs.
o f ( x) (Integrand): The function whose rate we are accumulating.
o dx (Differential): Represents the infinitely small width ( Δ x ) of the
rectangles.
Cumulative Effect of a Function
(Accumulation and the Definite Integral)
A CSUN Students Insight on Mastering Accumulation
As a Math 150A student transitioning into integrals, this topic—"The Cumulative
Effect of a Function"—is arguably the single most important conceptual leap in
calculus. Its the moment we move from instantaneous rates of change
(derivatives) to total change (integrals). Think of it this way: The derivative f ′ ( x)
tells you how fast your cars position is changing right now (velocity). The integral
b
∫ f ′ ( x) dx tells you the total distance the car traveled between time a and time b.
a
The whole idea boils down to: Area under a rate function equals total
accumulated quantity. We approximate this area using the sum of small rectangles
(Riemann Sums), and then we take the limit as the rectangles become infinitely thin
—thats the magic of the definite integral. Master the Fundamental Theorem of
Calculus (FTC), and you master accumulation.
I. Foundational Concepts: What is Accumulation?
A. The Problem of Area
Before calculus, to find the area of a shape, we needed a formula (e.g.,
A=length × width ). The concept of accumulation begins with the problem of finding
the area under a non-linear curve, f ( x), between two points, x=a and x=b .
When f (x) represents a rate of change (e.g., velocity in miles per hour), the area
under the curve represents the total change or accumulation of the original
quantity (e.g., total distance in miles).
B. Riemann Sums: The Accumulation Approximation
Since we cant find the area exactly with simple geometry, we approximate it using
rectangles. This process is called a Riemann Sum.
1. Partitioning the Interval: Divide the interval [a , b] into n subintervals of
equal width, Δ x .
, b−a
Δ x=
n
2. Choosing Sample Points: Within each subinterval, [ xi − 1 , x i] , we choose a
sample point x ∗i . The height of the i -th rectangle is determined by the
function value at this point, f ( x ∗i ).
3. Calculating the Area of Rectangles: The area of the i -th rectangle is
∗
Ai=f (x i ) ⋅ Δ x .
4. Summing the Areas: The total approximate area, Rn, is the sum of the areas
of all n rectangles:
n
Rn =∑ f ( x∗i ) ⋅ Δ x
i=1
The common choices for x ∗i are:
Left Endpoint Sum ( Ln): x ∗i =xi − 1
Right Endpoint Sum ( Rn ): x i =xi
∗
x i −1 + x i
Midpoint Sum ( M n): x ∗i =
2
C. The Definite Integral: The Limit of Accumulation
To move from the approximation (Riemann Sum) to the exact cumulative effect, we
must let the number of rectangles (n ) approach infinity, which simultaneously
forces the width of the rectangles ( Δ x ) to approach zero.
The Definite Integral is formally defined as the limit of the Riemann Sum:
b n
∫ f (x )dx= n→
lim ∑ f (x ∗i )⋅ Δ x
∞
a i =1
Notation Breakdown:
o ∫ (The Integral Sign): An elongated "S" representing "Sum" (the limit
of the summation).
o a and b (Limits of Integration): Define the interval over which
accumulation occurs.
o f ( x) (Integrand): The function whose rate we are accumulating.
o dx (Differential): Represents the infinitely small width ( Δ x ) of the
rectangles.
