Mathematics.

integral calculus

Integral

Calculus II50 minDifficulty5 out of 10

You should know: limit, continuity

Overview

The definite integral computes the net signed area between a function's graph and the x-axis over an interval. It's built by slicing the region into thin rectangles, summing their areas, and taking the limit as the rectangles become infinitely thin — the Riemann sum construction. The integral is the second pillar of calculus, and the Fundamental Theorem of Calculus reveals it to be intimately connected to the derivative: they are, in a precise sense, inverse operations.

Intuition

To find the area under a curve, chop the region into thin vertical strips, approximate each strip as a rectangle (width × height), and add up all the rectangle areas. That sum is only an approximation — but the thinner you make the strips, the closer the approximation gets to the true area. The integral is what that sum converges to as the strip width shrinks toward zero and the number of strips grows toward infinity.

Interactive Graph

Shade the area under the curve and watch the Riemann sum converge

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Formal Definition

Definition

The definite integral is defined as the limit of a Riemann sum:

∫abf(x) dx=lim⁡n→∞∑i=1nf(xi∗)Δx\int_a^b f(x)\,dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*) \Delta x

Δx = (b-a)/n is the width of each of n subintervals; x_i* is a sample point in the i-th subinterval

Riemann sum definition

Notation

NotationMeaning
∫\intIntegral sign, an elongated 'S' for 'sum', introduced by Leibniz
∫abf(x) dx\int_a^b f(x)\,dxThe definite integral of f from a to b
∫f(x) dx\int f(x)\,dxThe indefinite integral — the family of antiderivatives of f, plus a constant C
dxdxThe infinitesimal width of each slice, integrating with respect to x

Properties

Linearity

∫ab[cf(x)+dg(x)] dx=c∫abf(x) dx+d∫abg(x) dx\int_a^b [cf(x)+dg(x)]\,dx = c\int_a^b f(x)\,dx + d\int_a^b g(x)\,dx

Additivity over intervals

∫abf(x) dx+∫bcf(x) dx=∫acf(x) dx\int_a^b f(x)\,dx + \int_b^c f(x)\,dx = \int_a^c f(x)\,dx

Reversing limits negates

∫abf(x) dx=−∫baf(x) dx\int_a^b f(x)\,dx = -\int_b^a f(x)\,dx

Zero-width interval

∫aaf(x) dx=0\int_a^a f(x)\,dx = 0

Applications

Work done by a variable force, total distance traveled from a velocity function, and center of mass are all computed via integration.

Worked Examples

  1. The region under y=x from 0 to 2 is a right triangle with base 2 and height 2.

    Area=12(2)(2)=2\text{Area} = \frac{1}{2}(2)(2) = 2

Answer: 2

Practice Problems

Difficulty 4/10

Evaluate ∫₁³ (2x + 1) dx.

Difficulty 5/10

A vehicle's velocity is v(t) = 3t² m/s. How far does it travel between t = 0 and t = 4 s?

Difficulty 6/10

Hydrostatic pressure on a vertical dam face increases with depth as p(y) = ρg·y. For unit width, the total force is ∫₀ᴴ ρg·y dy. Find the force per unit width on a dam of height H = 10 m (use ρg = 9810 N/m³).

Common Mistakes

Common Mistake

Forgetting the constant of integration +C for indefinite integrals.

Any antiderivative family differs by a constant — omitting +C loses that entire family of solutions.

Common Mistake

Treating the integral sign and dx as decorative rather than meaningful notation.

dx specifies the variable of integration — essential in multivariable calculus where you might integrate the same expression with respect to different variables.

Quiz

What does the definite integral ∫ₐᵇ f(x) dx represent geometrically?

Flashcards

1 / 2

Historical Background

The method of computing areas by exhaustion goes back to Eudoxus (4th century BCE) and Archimedes, who computed the area of a parabolic segment using an early limiting process around 250 BCE. Newton and Leibniz unified area computation with the tangent-line problem in the 1670s, showing the two were inverse operations. Bernhard Riemann gave the modern rigorous definition of the definite integral as a limit of sums in 1854, the version most calculus courses still teach today.

  1. c. 250 BCE

    Archimedes computes areas via the method of exhaustion

    Archimedes

  2. 1670s

    Newton and Leibniz recognize integration and differentiation as inverse operations

    Isaac Newton, Gottfried Wilhelm Leibniz

  3. 1854

    Riemann formalizes the definite integral as a limit of sums

    Bernhard Riemann

Summary

  • The definite integral ∫ₐᵇ f(x)dx computes net signed area under a curve, defined as the limit of a Riemann sum.
  • The indefinite integral ∫f(x)dx is the family of antiderivatives of f, written with a +C.
  • Linearity and additivity over intervals make integrals easy to break apart and combine.
  • Integration and differentiation are inverse operations (formalized by the Fundamental Theorem of Calculus).
  • Applications range from physics (work, distance) to finance (present value) to ML (expectations).

References

  1. BookRiemann, B. (1854). Über die Darstellbarkeit einer Function durch eine trigonometrische Reihe.
  2. WebsiteWikipedia — Integral