Riemann Sum( 黎曼和): An approximation of the definite integral (and area under a curve) calculated by dividing the area into rectangles, finding the area of each rectangle, and summing the areas.
Partition (分割): The act of dividing something into parts.
Lower Sum (下和): A Riemann sum where the height of each rectangle is determined by the minimum value of the function on that subinterval.
Upper Sum (上和): A Riemann sum where the height of each rectangle is determined by the maximum value of the function on that subinterval.
Integrand (被積函數): The function being integrated in a definite integral.
Definite Integral (定積分): The limit of Riemann sums as the width of the subintervals approaches zero.
Indefinite Integral (不定積分): An integral that does not have specific limits of integration. Represented by the symbol ∫f(x) dx.
Improper Integrals (瑕積分): The interval of integration is infinite, or the integrand has a singularity (a point where it becomes undefined or infinite).
Fundamental Theorem of Calculus (微積分基本定理): Relates differentiation and integration, providing a way to calculate definite integrals using antiderivatives.
Antiderivative (反導數): A function whose derivative is the given function.
Substitution (積分代換): A technique for evaluating integrals by making a change of variables to simplify the integrand.
Integration with Respect to x (對 x 積分): Using vertical rectangles to approximate the area, leading to an integral in terms of x.
Integration with Respect to y (對 y 積分): Using horizontal rectangles to approximate the area, leading to an integral in terms of y.
Region (區域): A part of a space or surface enclosed within a boundary.
Area Between Curves (曲線間面積): The area enclosed between two curves and two vertical lines.
Integration by Parts (分部積分法): A technique for simplifying integrals of a product of functions.
Trigonometric Substitution (三角代換法): Replace the variable in the integrand with a trigonometric function.
Partial Fractions (部分分式法): Break down a complex rational function into simpler fractions.
Irreducible (不可約的): Not able to be factored.
Proper (真分式): A fraction where the degree of the numerator is less than the degree of the denominator.