Russian Translation(s) & Details for 'integral'

English Word: integral

Key Russian Translations:

• интеграл [ɪnʲtʲɪˈɡral] - [Formal, Mathematical term; used in scientific and educational contexts]

Frequency: Medium (Common in mathematical, scientific, and academic discussions, but less frequent in everyday conversation)

Difficulty: B2 (Intermediate; requires familiarity with mathematical terminology and basic Russian noun inflections)

Pronunciation (Russian):

интеграл: [ɪnʲtʲɪˈɡral]

Note on интеграл: The stress falls on the third syllable ("ɡral"). Be mindful of the soft sign (ʲ) which softens the preceding consonant, a common feature in Russian pronunciation. Pronunciation may vary slightly by regional dialect.

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Meanings and Usage:

Primary Meaning: A mathematical concept referring to the result of integration, or the process of finding the area under a curve.

Translation(s) & Context:

• интеграл - Used in formal mathematical contexts, such as calculus and physics, to denote the antiderivative or definite integral.

Usage Examples:

• Этот интеграл решается с помощью тригонометрических подстановок.

  This integral is solved using trigonometric substitutions.

• В курсе анализа мы изучили определённый интеграл и его применение в физике.

  In the analysis course, we studied the definite integral and its applications in physics.

• Интеграл от функции f(x) равен F(x) + C, где C — константа.

  The integral of the function f(x) is equal to F(x) + C, where C is a constant.

• Решение двойного интеграла требует понимания координатных систем.

  Solving a double integral requires understanding of coordinate systems.

• В этой формуле интеграл обозначает суммирование площади под кривой.

  In this formula, the integral represents the summation of the area under the curve.

Secondary Meaning: As an adjective, meaning essential or necessary (less common translation for 'integral' in English).

Translation(s) & Context:

• необходимый - Used in general contexts to mean something that is indispensable or integral to a whole.

Usage Examples:

• Эта часть необходима для целостности системы.

  This part is integral/necessary for the integrity of the system.

• В демократическом обществе свобода слова — необходимый элемент.

  In a democratic society, freedom of speech is an integral/necessary element.

Russian Forms/Inflections:

The primary translation, "интеграл", is a masculine noun in Russian. It follows the standard third-declension pattern for masculine nouns ending in a consonant. Below is a table outlining its key inflections:

For the secondary translation "необходимый", it is an adjective that agrees in gender, number, and case with the noun it modifies. It has regular adjective inflections (e.g., необходимая in feminine singular, необходимые in plural).

Russian Synonyms/Antonyms:

• Synonyms:
  • интеграция (integration; more about the process)
  • суммирование (summation; in a broader mathematical sense, but with subtle differences in precision)
• Antonyms:
  • дифференциал (differential; the opposite concept in calculus)

Related Phrases:

• определённый интеграл - Definite integral; used when specifying limits in mathematical calculations.
• неопределённый интеграл - Indefinite integral; refers to the family of antiderivatives.
• вычислить интеграл - To compute the integral; a common phrase in problem-solving contexts.

Usage Notes:

In Russian, "интеграл" directly corresponds to the English mathematical term and is used almost identically in technical contexts. However, for the adjectival sense of "integral" (e.g., essential), "необходимый" is a more precise match than "интеграл". Choose based on context: use "интеграл" for math and "необходимый" for general descriptions. Note that Russian adjectives like "необходимый" must agree in gender, number, and case with the noun they describe, which is a key grammatical difference from English. This word is common in formal writing or lectures but rare in casual speech.

Common Errors:

• Mistake: Confusing "интеграл" with "интеграция" (integration as a process). Incorrect: Использовать интеграл для процесса. Correct: Использовать интеграцию для процесса. Explanation: "Интеграл" is the result, not the action; this mix-up often occurs among beginners learning calculus terminology.
• Mistake: Failing to inflect "интеграл" correctly in sentences. Incorrect: Я изучаю интеграл (in genitive context). Correct: Я изучаю интеграла (genitive form). Explanation: Russian nouns change based on case, so learners must pay attention to sentence structure to avoid grammatical errors.

Cultural Notes:

The concept of "интеграл" is deeply rooted in Russian mathematical tradition, influenced by pioneers like Leonhard Euler and Soviet-era advancements in physics and engineering. In Russian culture, mathematics is highly valued in education, and terms like this are integral to STEM fields, reflecting Russia's historical emphasis on science during the Space Race era.

Related Concepts:

• дифференциал
• производная
• функция