Essence of calculus
1. Fundamental Theorem of Calculus: The Integral
Fundamental Theorem of Calculus: The Integral
Antiderivative and Differentiation
Antiderivative: A function $F(x)$ is an antiderivative of a function
$f(x)$ if $F’(x) = f(x)$.
The process of finding an antiderivative is called anti-differentiation
or integration.
Constant Function and its Role in Integration
Any two antiderivatives of a function differ by a constant.
This constant is called the constant of integration.
Notation: $\int f(x) dx = F(x) + C$, where $F’(x) = f(x)$ and $C$ is
the constant of integration.
Signed Area and its Relation to Integration
The integral of a non-negative function gives the area between the
graph of the function and the x-axis.
If the function takes on both positive and negative values, then the
integral gives the signed area, which is the area above the x-axis
minus the area below the x-axis.
The Fundamental Theorem of Calculus provides a way to compute
definite integrals (i.e., integrals over a specific interval) using
antiderivatives.
2. Antiderivative and Differentiation
Y = f(x) and the x-axis over an interval [a, b] is given by the definite
integral ∫[a, b] f(x) dx. The definite integral gives us the net signed
area between the curve and the x-axis over the interval [a, b]. This
is because the definite integral is defined as the difference between
the antiderivative evaluated at the upper limit and the
antiderivative evaluated at the lower limit: ∫[a, b] f(x) dx = F(b) –
F(a).
The signed area is called “signed” because it can be positive or
negative, depending on the orientation of the curve with respect to
the x-axis. If the curve is above the x-axis, the signed area is
positive, and if the curve is below the x-axis, the signed area is
1. Fundamental Theorem of Calculus: The Integral
Fundamental Theorem of Calculus: The Integral
Antiderivative and Differentiation
Antiderivative: A function $F(x)$ is an antiderivative of a function
$f(x)$ if $F’(x) = f(x)$.
The process of finding an antiderivative is called anti-differentiation
or integration.
Constant Function and its Role in Integration
Any two antiderivatives of a function differ by a constant.
This constant is called the constant of integration.
Notation: $\int f(x) dx = F(x) + C$, where $F’(x) = f(x)$ and $C$ is
the constant of integration.
Signed Area and its Relation to Integration
The integral of a non-negative function gives the area between the
graph of the function and the x-axis.
If the function takes on both positive and negative values, then the
integral gives the signed area, which is the area above the x-axis
minus the area below the x-axis.
The Fundamental Theorem of Calculus provides a way to compute
definite integrals (i.e., integrals over a specific interval) using
antiderivatives.
2. Antiderivative and Differentiation
Y = f(x) and the x-axis over an interval [a, b] is given by the definite
integral ∫[a, b] f(x) dx. The definite integral gives us the net signed
area between the curve and the x-axis over the interval [a, b]. This
is because the definite integral is defined as the difference between
the antiderivative evaluated at the upper limit and the
antiderivative evaluated at the lower limit: ∫[a, b] f(x) dx = F(b) –
F(a).
The signed area is called “signed” because it can be positive or
negative, depending on the orientation of the curve with respect to
the x-axis. If the curve is above the x-axis, the signed area is
positive, and if the curve is below the x-axis, the signed area is
