Calculus

The Definite Integral

During the plague years of 1665-1666, Newton worked out his method of fluxions in isolation in Woolsthorpe. A decade later Leibniz reached the same idea independently and published it in 1684. They had found the same fact: area accumulation and tangent slopes are inverse operations. Two thousand years of Archimedes' geometry collapsed into one formula.

**Simpson's rule in SciPy**: scipy.integrate.simpson approximates a definite integral over 3 points with error $O(h^4)$. Used whenever the antiderivative is not elementary
**Black-Scholes (1973)**: a European option price is a definite integral of a log-normal density. Every day, markets reprice trillions of dollars through this formula
**Moments of inertia in SolidWorks**: $\int r^2\,dm$ over the part volume defines the moment of inertia of a crankshaft, a turbine blade, or any rotating structure
**LIGO (2015)**: gravitational-wave detection is a correlation integral $\int h(t)\,s(t)\,dt$ of signal against template. Sensitivity is set by the convergence of that integral

Предварительные знания

Antiderivatives and the indefinite integral
The concept of a limit
Sums and sigma notation

Riemann Sums: area as a limit

We want the area under $f(x)$ on $[a, b]$. The plan: partition the interval into $n$ pieces, erect a rectangle over each piece, add the areas, and send $n \to \infty$.

Type	Formula	Height used
Left $L_n$	$\sum_{i=0}^{n-1} f(x_i)\Delta x$	Left endpoint of each subinterval
Right $R_n$	$\sum_{i=1}^{n} f(x_i)\Delta x$	Right endpoint of each subinterval
Midpoint $M_n$	$\sum_{i=1}^{n} f\left(\frac{x_{i-1}+x_i}{2}\right)\Delta x$	Midpoint of each subinterval

As $n \to \infty$, all three sums converge to the same limit - the **definite integral** (provided the function is integrable).

**Key distinction**: the indefinite integral $\int f(x)\,dx$ is a family of functions ($+ C$), while the definite integral $\int_a^b f(x)\,dx$ is a **number**!

When computing $\int_0^2 x\,dx$ using a right Riemann sum with $n = 4$, what is $\Delta x$?

Properties and geometric meaning

When $f(x) \geq 0$ on $[a, b]$, the integral equals the **area of the curvilinear trapezoid** bounded by the graph $y = f(x)$, the $x$-axis, and the verticals $x = a$, $x = b$.

When the function is negative (graph below the $x$-axis), the integral is **negative** - area with a minus sign. $\int_0^\pi \sin x\,dx = 2$, $\int_\pi^{2\pi} \sin x\,dx = -2$, $\int_0^{2\pi} \sin x\,dx = 0$.

**Five core properties:**

**Mean Value Theorem for integrals**: if $f$ is continuous on $[a, b]$, there exists $c \in [a, b]$ such that $\int_a^b f(x)\,dx = f(c)\cdot(b - a)$. Solving for $f(c)$ gives the average value:

The integral always equals the geometric area of the region

The integral gives the signed area: positive above the axis, negative below

For geometric area (always positive), split the region: $S = \int_a^c f(x)\,dx + \left|\int_c^b f(x)\,dx\right|$

If $\int_0^4 f(x)\,dx = 20$, what is the average value of $f$ on $[0, 4]$?

The Fundamental Theorem of Calculus

During the plague years of 1665-1666, Newton developed his method of fluxions in Woolsthorpe and saw that area accumulation and tangent slopes are inverse operations. Leibniz arrived at the same insight independently around 1675-1676 and published it in 1684 in Acta Eruditorum. Two thousand years of Archimedes' geometry collapsed into a single formula.

**Why this matters**: before the Fundamental Theorem, area accumulation (integration) and tangent slopes (differentiation) seemed completely unrelated. The theorem showed they are inverse operations - like multiplication and division.

Example: $\int_1^3 x^2\,dx = \frac{x^3}{3}\Big|_1^3 = \frac{27}{3} - \frac{1}{3} = \frac{26}{3} \approx 8.67$. No summing - just find an antiderivative and plug in the limits.

Using the Newton-Leibniz formula, what is $\int_0^4 2x\,dx$?

Summary

**Definite integral** $\int_a^b f(x)\,dx$ - limit of Riemann sums, a number
**Geometric meaning**: signed area under the curve
**Properties**: linearity, additivity, reversing limits changes the sign
**Mean Value Theorem**: $\int_a^b f\,dx = f(c)(b-a)$ for some $c \in [a,b]$
**Fundamental Theorem**: $\int_a^b f(x)\,dx = F(b) - F(a)$ - differentiation and integration are inverse operations

The Fundamental Theorem of Calculus

The definite integral bridges two great ideas:

Newton-Leibniz Formula — Computing integrals via antiderivatives - next lesson!
Integration Techniques — Methods for complex functions
Improper Integrals — Integrals with infinite limits

Вопросы для размышления

Why is the integral negative when the function lies below the $x$-axis, rather than zero?
How are the left and right Riemann sums related for a monotone function?
Why does reversing the limits of integration change the sign?
What is the geometric meaning of the Mean Value Theorem for integrals?

Связанные уроки

Calculus

The Definite Integral

**Simpson's rule in SciPy**: scipy.integrate.simpson approximates a definite integral over 3 points with error $O(h^4)$. Used whenever the antiderivative is not elementary
**Black-Scholes (1973)**: a European option price is a definite integral of a log-normal density. Every day, markets reprice trillions of dollars through this formula
**Moments of inertia in SolidWorks**: $\int r^2\,dm$ over the part volume defines the moment of inertia of a crankshaft, a turbine blade, or any rotating structure
**LIGO (2015)**: gravitational-wave detection is a correlation integral $\int h(t)\,s(t)\,dt$ of signal against template. Sensitivity is set by the convergence of that integral

Предварительные знания

Antiderivatives and the indefinite integral
The concept of a limit
Sums and sigma notation

Riemann Sums: area as a limit

We want the area under $f(x)$ on $[a, b]$. The plan: partition the interval into $n$ pieces, erect a rectangle over each piece, add the areas, and send $n \to \infty$.

Type	Formula	Height used
Left $L_n$	$\sum_{i=0}^{n-1} f(x_i)\Delta x$	Left endpoint of each subinterval
Right $R_n$	$\sum_{i=1}^{n} f(x_i)\Delta x$	Right endpoint of each subinterval
Midpoint $M_n$	$\sum_{i=1}^{n} f\left(\frac{x_{i-1}+x_i}{2}\right)\Delta x$	Midpoint of each subinterval

As $n \to \infty$, all three sums converge to the same limit - the **definite integral** (provided the function is integrable).

**Key distinction**: the indefinite integral $\int f(x)\,dx$ is a family of functions ($+ C$), while the definite integral $\int_a^b f(x)\,dx$ is a **number**!

When computing $\int_0^2 x\,dx$ using a right Riemann sum with $n = 4$, what is $\Delta x$?

Properties and geometric meaning

When $f(x) \geq 0$ on $[a, b]$, the integral equals the **area of the curvilinear trapezoid** bounded by the graph $y = f(x)$, the $x$-axis, and the verticals $x = a$, $x = b$.

**Five core properties:**

**Mean Value Theorem for integrals**: if $f$ is continuous on $[a, b]$, there exists $c \in [a, b]$ such that $\int_a^b f(x)\,dx = f(c)\cdot(b - a)$. Solving for $f(c)$ gives the average value:

The integral always equals the geometric area of the region

The integral gives the signed area: positive above the axis, negative below

For geometric area (always positive), split the region: $S = \int_a^c f(x)\,dx + \left|\int_c^b f(x)\,dx\right|$

If $\int_0^4 f(x)\,dx = 20$, what is the average value of $f$ on $[0, 4]$?

The Fundamental Theorem of Calculus

Example: $\int_1^3 x^2\,dx = \frac{x^3}{3}\Big|_1^3 = \frac{27}{3} - \frac{1}{3} = \frac{26}{3} \approx 8.67$. No summing - just find an antiderivative and plug in the limits.

Using the Newton-Leibniz formula, what is $\int_0^4 2x\,dx$?

Summary

**Definite integral** $\int_a^b f(x)\,dx$ - limit of Riemann sums, a number
**Geometric meaning**: signed area under the curve
**Properties**: linearity, additivity, reversing limits changes the sign
**Mean Value Theorem**: $\int_a^b f\,dx = f(c)(b-a)$ for some $c \in [a,b]$
**Fundamental Theorem**: $\int_a^b f(x)\,dx = F(b) - F(a)$ - differentiation and integration are inverse operations

The Fundamental Theorem of Calculus

The definite integral bridges two great ideas:

Newton-Leibniz Formula — Computing integrals via antiderivatives - next lesson!
Integration Techniques — Methods for complex functions
Improper Integrals — Integrals with infinite limits

Вопросы для размышления

Why is the integral negative when the function lies below the $x$-axis, rather than zero?
How are the left and right Riemann sums related for a monotone function?
Why does reversing the limits of integration change the sign?
What is the geometric meaning of the Mean Value Theorem for integrals?