integral
/ˈɪntɪɡrəl/ (bre, ipa) · /ˈɪntɪɡrəl/ (ame, ipa) · /ˈin-ti-grəl (usually so in mathematics) in-ˈte-grəl also -ˈtē- also nonstandard ˈin-trə-gəl/ (ame, mw) · /ˈɪn.tɪ.ɡrəl/ (bre, ipa) · /ˈɪn.t̬ə.ɡrəl/ (ame, ipa)
integral — 形容詞
- integralpositive
- more integralcomparative
- most integralsuperlative
1. belonging to something in such a central way that the whole thing cannot functio
不可或缺的
構成整體所必需的
belonging to something in such a central way that the whole thing cannot function, exist, or be complete without it
The software is now an integral part of the school's daily teaching system.
這套軟體現在已是學校日常教學系統中不可或缺的一部分。
collocation: integral part of
Good communication skills were integral to Asher's success as a team leader.
良好的溝通能力是 Asher 成功擔任團隊領導者的關鍵。
pattern: integral + to + noun
Exercise and a healthy diet are integral parts of a strong recovery plan.
運動與健康飲食是強力復原計畫中不可或缺的部分。
The new public library became integral to the life of the local community.
這座新的公共圖書館成了當地社區生活中不可或缺的一部分。
Trust is integral to any strong friendship between two people.
信任是任何牢固友誼中不可或缺的要素。
- essential
the closest synonym; both suggest something cannot be removed without damaging the whole
- crucial
emphasises importance for success or outcome; slightly stronger than 'integral'
- fundamental
stresses being at the base or foundation of something
- vital
implies something is needed for life or continued existence; stronger than 'integral'
- peripheral
describes something that is on the edge and not central
- inessential
means something can be removed without affecting the whole
文法句型
integral + to + noun phrase
integral + part of + noun phrase
用法筆記
Often followed by 'to' to show what something is essential for. Can also appear before 'part of' — 'an integral part of something' is a common fixed expression.
常見錯誤
2. designed or built into something as one unit, rather than being added on or supp
內置的
構成一體的,非外加的
designed or built into something as one unit, rather than being added on or supplied as a separate item
The new car has an integral GPS system built right into the dashboard.
這輛新車的儀表板內建了 GPS 系統。
collocation: integral + noun (built-in feature)
In this design, the handle is integral to the frame, so you cannot remove it.
在這個設計中,把手與框架是一體的,因此無法拆下。
pattern: integral to + noun
The speakers are integral to the television, not separate add-on devices.
這些揚聲器是電視的內置元件,不是外接裝置。
Takeshi's kitchen has an integral waste-disposal unit under the sink.
Takeshi 的廚房水槽下方裝有內置式廢棄物處理裝置。
Modern smartphones have an integral camera that cannot be detached.
現代智慧型手機的相機是內建的,無法拆卸。
- built-in
more common in everyday speech; 'integral' is slightly more formal
- inbuilt
British English variant; same meaning as 'built-in'
- incorporated
suggests something has been deliberately included as part of a design
- separate
describes parts that are not physically joined and can be removed
- detachable
means designed to be taken off or disconnected
- external
means located outside the main body or unit
文法句型
integral + noun
integral + to + noun phrase
用法筆記
Commonly describes parts or features of machines, devices, or buildings that are made as one piece rather than added later. Distinguish from sense 1 (ESSENTIAL PART), where the focus is on importance; here the focus is on physical or structural unity.
常見錯誤
3. containing every part that is needed, with nothing missing; complete and entire
完整的
各部分齊全無缺的
containing every part that is needed, with nothing missing; complete and entire
The museum proudly presented its integral pottery collection after finally acquiring the missing Ming vase.
博物館在終於取得那件遺失的明代花瓶後,驕傲地展出了完整的陶器收藏。
attributive: integral + noun for complete collection
The historian managed to reconstruct the integral document from pieces found in three different archives.
這位歷史學家設法從三個不同檔案館找到的碎片中,重建出完整的文件。
After finding the last rare issue, Élise finally owned an integral stamp collection spanning every period.
Élise 在找到最後那張珍稀郵票後,終於擁有了一套涵蓋每個時期的完整郵票收藏。
Adina's integral set of antique coins now includes every year from the series without exception.
Adina 的完整古幣套組如今包含了該系列中的每一個年份。
The orchestra performed the integral symphony, including all four movements.
樂團演奏了完整的交響曲,包含全部四個樂章。
- incomplete
describes something that is missing one or more parts
- partial
means only part of the whole is present
文法句型
integral + noun
become + integral
be + integral
用法筆記
This sense is formal and less frequent than senses 1 and 2. It is often used in academic writing, art criticism, or historical descriptions to emphasise that a collection or text has all its parts. Distinguish from sense 2 (BUILT-IN), which describes parts joined together physically; sense 3 describes a whole thing whose all parts are present.
integral — 名詞
- integralsingular
- integralsplural
1. a number or formula that represents the total value of a changing quantity over
積分
微積分中求曲線下方面積的函數
a number or formula that represents the total value of a changing quantity over a certain range, especially used in calculus to calculate areas under curves and volumes of solids
In calculus class, the students learned how to calculate the integral of a simple quadratic function.
在微積分課上,學生們學會了如何計算簡單二次函數的積分。
collocation: calculate the integral
The definite integral of the function between x = 1 and x = 4 gives the area under the curve.
該函數在 x = 1 與 x = 4 之間的定積分,即為曲線下的面積。
collocation: definite integral
Adina used a computer program to find the integral of the equation.
Adina 用電腦程式來求該方程式的積分。
To find the volume of the solid, the engineer first had to solve a double integral.
為了求出這個立體的體積,工程師必須先解一個二重積分。
The integral of velocity over time equals the total distance travelled.
速度對時間的積分等於行駛的總距離。
- antiderivative
technical synonym for indefinite integral; less common in basic calculus courses
- definite integral
a specific type of integral that produces a numerical value over a defined range
- derivative
the inverse operation in calculus; a derivative measures the rate of change, whereas an integral measures accumulation
文法句型
the + integral + of + noun phrase
definite/indefinite + integral
calculate/find/solve + the + integral
用法筆記
A technical term in calculus. The two main types are the 'definite integral' (which gives a number) and the 'indefinite integral' (which gives a family of functions). Not to be confused with 'integer', which is a whole number.