3.7 Integration: Glossary

Costas Loizou; Shuwei Chen; Deak Kirkham

3.7 Integration: Glossary

The pronunciation guide given in these glossaries draws on the symbols International Phonetic Alphabet. We have made two adaptations to the IPA system. Firstly, we use slanted lines as opposed to square brackets to enclose pronunciation material. We do this as the slanted line convention is more common in language teaching contexts. Secondly, we use the symbol <r> to denote the voiced alveolar approximant sound which appears in British English at the beginning of the each word in the phrase 'red rhombus'; the IPA would instead use an inverted 'r' symbols namely <ɹ>. We have done this to avoid unnecessary complication to the visuals of the glossaries.

In addition to these conventions, please also note that the pronunciations given are indicative of contemporary Standard British English pronunciation and should not be read as normative, nor as a target pronunciation that you must aim to produce. The pronunciation of English varies widely across World Englishes; part of becoming fluent in the academic English of mathematics is developing an awareness of, and sensitivity to, the variation in pronunciation that you will encounter.

Word	Pronunciation	Part of speech词性	Related terminology	Chinese meaning	Example sentence 1	Example sentence 2
integration	/ˌɪntɪˈɡreɪʃən/	n.	integral n. 积分（的结果），积分表达式； integrate v. 对......进行积分	积分（的过程），积分运算	The area under the curve can be found by integration over \([0, 2]\). 曲线下的面积可以通过在 \([0,2]\) 上的积分求出。	The integration of \(2x\) with respect to \(x\) is \(x^2+C\). 对 \(2x\) 关于 \(x\) 进行积分得到 \(x^2+C\)。
integrand	/ˈɪntɪɡrænd/	n.		被积函数	the integrand of the integral 积分的被积函数	In the integral \(\int x^2 dx\), the integrand is \(x^2\). 在积分 \(\int x^2 dx\) 中，被积函数是 \(x^2\)。
indefinite integral	/ɪnˈdefɪnət ˈɪntɪɡrəl/	n. phr.	definite integral 定积分	不定积分	The indefinite integral of \(2x\) is \(x^2 + C\). \(2x\) 的不定积分是 \(x^2 + C\)。	The indefinite integral represents the general antiderivative of a function. 不定积分表示函数的一般原函数。
continuous	/kənˈtɪnjuəs/	adj.		连续的	The function is continuous on the interval \([0,1]\). 该函数在区间 \([0,1]\) 上连续。	continuous at a point 在某一点处连续
proper	/ˈprɒpə/	adj.	proper rational function 真有理函数，真分式	适当的	This is a proper integral because the limits are finite. 这是一个正常积分，因为积分上下限是有限的。	Before applying partial fraction decomposition, we must ensure the expression is a proper rational function. 在进行部分分式分解之前，我们必须确保该表达式是真有理函数。
improper	/ɪmˈprɒpə/	adj.	Improper rational Function 假有理函数，假分式	反常的	The integral is improper because the upper limit is infinity. 该积分是反常积分，因为上限是无穷大。	We divide the polynomials to express the improper rational function as a sum of a polynomial and a proper rational function. 我们通过多项式除法，将假有理函数表示为一个多项式与一个真有理函数之和。
antiderivative	/ˌæntiˈdɪərɪvətɪv/	n.		原函数	\(F(x)=x^2\) is an antiderivative of \(f(x)=2x\). \(F(x)=x^2\) 是\(f(x)=2x\)的一个原函数。	Finding an antiderivative is equivalent to computing an indefinite integral. 求原函数等价于计算不定积分。
dummy variable	/ˈdʌmi ˈveəriəbl/	n. phr.		虚拟变量，哑变量	We can change the dummy variable from \(x\) to \(t\). 我们可以把哑变量从 \(x\) 改为 \(t\)。	The result does not depend on the choice of dummy variable. 结果不依赖于哑变量的选择。
irreducible	/ˌɪrɪˈdjuːsəbəl/	adj.		不可约的	The rational function can be decomposed into partial fractions with irreducible denominators. 这个有理函数可以分解为分母不可约的部分分式。	The denominator \(x^2+1\) is irreducible over the real numbers, so we use a trigonometric substitution. 分母\(x^2+1\) 在实数范围内不可约，因此我们使用三角代换。
decomposition	/ˌdiːˌkɒmpəˈzɪʃən/	n.	partial fraction decomposition 部分分式分解	分解	We use partial fraction decomposition to simplify the integral. 我们用部分分式分解来简化积分。	The function admits a decomposition into two simpler functions. 该函数可以分解为两个更简单的函数。
integral symbol	/ˈɪntɪɡrəl ˈsɪmbəl/	n. phr.		积分符号	We interpret the integral symbol as a limit of sums. 我们将积分符号理解为求和的极限。	The double integral symbol is used for area. 二重积分符号用于面积计算。
limits of integration	/ˈlɪmɪts əv ˌɪntɪˈɡreɪʃən/	n. phr.		积分上下限	upper limit of integration 积分上限	lower limit of integration 积分下限
fundamental theorem of calculus	/ˌfʌndəˈmentl ˈθɪərəm əv ˈkælkjʊləs/	n. phr.	FTC	微积分基本定理	We apply the Fundamental Theorem of Calculus to evaluate the integral. 我们用微积分基本定理计算积分。	This result follows from the Fundamental Theorem of Calculus. 该结果由微积分基本定理推出。
rules of integration	/ruːlz əv ˌɪntɪˈɡreɪʃən/	n. phr.		积分法则	The rules of integration are derived from derivatives. 积分法则来源于导数。	These are standard rules of integration. 这些是标准积分法则。
integration by parts	/ˌɪntɪˈɡreɪʃən baɪ pɑːts/	n. phr.	缩写 i.b.p.	分部积分法	We use integration by parts to evaluate the integral. 我们用分部积分法计算该积分。	formula for integration by parts 分部积分公式
integration by substitution	/ˌɪntɪˈɡreɪʃn baɪ ˌsʌbstɪˈtjuːʃn/	n. phr.		换元积分法	We use integration by substitution to simplify the integral. 我们使用换元积分法来简化这个积分。	Let \(u = g(x)\) be the substitution in the integration process. 令 \(u = g(x)\) 是积分过程中的换元。
trigonometric substitution	/ˌtrɪɡənəˈmetrɪk ˌsʌbstɪˈtjuːʃən/	n. phr.		三角代换	To evaluate \(\int \sqrt{1−x^2} dx\), we use the trigonometric substitution \(x =\ sin \theta\). 为了计算积分\(\int \sqrt{1−x^2} dx\)，我们使用三角代换 \(x = \sin \theta\)。	Apply trigonometric substitution for square roots. 对根式使用三角代换。