CSUN MATH 150A: The Cumulative Effect of a Function
(The Definite Integral)
I. Personal Learning Insights: The Core Idea
Welcome to the world of integration! As a Math 150A student, the concept of the
Cumulative Effect of a Function is the most significant philosophical shift from
differential calculus. While the derivative answers the question, "How fast is it
changing now?" the integral answers, "How much total change has occurred?"
My core insight is that integration is fundamentally about multiplication and
summation. If a function f (x) represents a rate (e.g., miles per hour), and we
multiply that rate by a tiny amount of time ( Δ t ), we get a tiny amount of distance
traveled ( f (t)⋅ Δt ). The integral is merely the tool that takes the limit of the sum of
these infinitely many tiny products. Area under the curve = Accumulation of the
rate. Internalizing this connection is more important than memorizing any specific
formula.
The major breakthrough, the Fundamental Theorem of Calculus (FTC), is the
realization that this complex process of summing infinite rectangles can be reduced
to simple subtraction ( F (b)− F( a)). This makes the entire field of accumulation
tractable and efficient.
II. Detailed Conceptual Review: From Approximation to Precision
A. The Genesis of Accumulation: The Area Problem
The formal study of accumulation begins with the geometric challenge of finding the
area under a curve y=f (x ) over an interval [a , b]. For a smooth curve, standard
geometric formulas fail. The solution is the method of exhaustion, formalized by the
Riemann Sum.
1. Partitioning the Interval: The interval [a , b] is divided into n equal
subintervals.
o The width of each subinterval is the change in x :
b−a
Δ x=
n
, ∗
2. Forming Rectangles: Within the i -th subinterval, [ xi − 1 , x i] , a sample point x i
∗
is chosen. The height of the rectangle is f ( x i ).
o The area of the i -th rectangle is Ai=f ( x ∗i ) ⋅ Δ x .
3. The Riemann Sum: The total approximate area ( Rn ) is the sum of all n
rectangular areas:
n
Rn =∑ f ( xi ) ⋅ Δ x
∗
i=1
B. The Definite Integral: The Limit of the Sum
The exact cumulative effect, or the area under the curve, is obtained by taking the
limit as the number of subintervals n approaches infinity (and thus Δ x →0 ).
The Definite Integral is defined as this limit:
b n
∫ f (x )dx= n→
lim ∑ f (x i )⋅ Δ x
∞
∗
a i =1
Integration Notation:
b
o ∫ ❑: The limits of integration specify the interval of accumulation.
a
o f ( x): The integrand, the function whose effect is being accumulated
(often a rate function).
o dx : The differential, representing the infinitesimal width of the
accumulation slices.
Conceptual Interpretation: If f ( x)≥ 0, the integral is the area under the curve. If
f ( x) represents a rate of change, the integral represents the net change in the
original quantity.
III. The Fundamental Theorem of Calculus (FTC)
The FTC is the mathematical "easy button" for accumulation, linking differentiation
and integration as inverse processes.
A. FTC Part I (The Accumulation Function)
The first part of the theorem shows that accumulation is a reversible process.
Statement: If f is continuous on [a , b], then the function g( x ) defined by:
(The Definite Integral)
I. Personal Learning Insights: The Core Idea
Welcome to the world of integration! As a Math 150A student, the concept of the
Cumulative Effect of a Function is the most significant philosophical shift from
differential calculus. While the derivative answers the question, "How fast is it
changing now?" the integral answers, "How much total change has occurred?"
My core insight is that integration is fundamentally about multiplication and
summation. If a function f (x) represents a rate (e.g., miles per hour), and we
multiply that rate by a tiny amount of time ( Δ t ), we get a tiny amount of distance
traveled ( f (t)⋅ Δt ). The integral is merely the tool that takes the limit of the sum of
these infinitely many tiny products. Area under the curve = Accumulation of the
rate. Internalizing this connection is more important than memorizing any specific
formula.
The major breakthrough, the Fundamental Theorem of Calculus (FTC), is the
realization that this complex process of summing infinite rectangles can be reduced
to simple subtraction ( F (b)− F( a)). This makes the entire field of accumulation
tractable and efficient.
II. Detailed Conceptual Review: From Approximation to Precision
A. The Genesis of Accumulation: The Area Problem
The formal study of accumulation begins with the geometric challenge of finding the
area under a curve y=f (x ) over an interval [a , b]. For a smooth curve, standard
geometric formulas fail. The solution is the method of exhaustion, formalized by the
Riemann Sum.
1. Partitioning the Interval: The interval [a , b] is divided into n equal
subintervals.
o The width of each subinterval is the change in x :
b−a
Δ x=
n
, ∗
2. Forming Rectangles: Within the i -th subinterval, [ xi − 1 , x i] , a sample point x i
∗
is chosen. The height of the rectangle is f ( x i ).
o The area of the i -th rectangle is Ai=f ( x ∗i ) ⋅ Δ x .
3. The Riemann Sum: The total approximate area ( Rn ) is the sum of all n
rectangular areas:
n
Rn =∑ f ( xi ) ⋅ Δ x
∗
i=1
B. The Definite Integral: The Limit of the Sum
The exact cumulative effect, or the area under the curve, is obtained by taking the
limit as the number of subintervals n approaches infinity (and thus Δ x →0 ).
The Definite Integral is defined as this limit:
b n
∫ f (x )dx= n→
lim ∑ f (x i )⋅ Δ x
∞
∗
a i =1
Integration Notation:
b
o ∫ ❑: The limits of integration specify the interval of accumulation.
a
o f ( x): The integrand, the function whose effect is being accumulated
(often a rate function).
o dx : The differential, representing the infinitesimal width of the
accumulation slices.
Conceptual Interpretation: If f ( x)≥ 0, the integral is the area under the curve. If
f ( x) represents a rate of change, the integral represents the net change in the
original quantity.
III. The Fundamental Theorem of Calculus (FTC)
The FTC is the mathematical "easy button" for accumulation, linking differentiation
and integration as inverse processes.
A. FTC Part I (The Accumulation Function)
The first part of the theorem shows that accumulation is a reversible process.
Statement: If f is continuous on [a , b], then the function g( x ) defined by:
