Calculus Volume 2 · 免费预览
定积分
定积分是黎曼和随着划分不断加密时的极限，它是一个与取样方式无关的数，并具有线性和可加性等性质，因而既便于计算，又作为带符号面积具有几何意义。
1. The limit definition
The limit definition
2. Properties from the definition
Properties from the definition
3. Evaluating from the definition
Evaluating from the definition
4. The mean value theorem for integrals
The mean value theorem for integrals
5. Integral as signed area
The definite integral of f from a to b measures what geometric quantity?
6. Additivity property
Which equation expresses the additivity of the definite integral over adjacent intervals?
7. Reversed limits
According to the convention for reversed limits, what is the integral from 3 to 1 of f(x) dx in terms of the integral from 1 to 3?
8. Comparison property
On [0, 1], the function f satisfies 0 at most f(x) at most 1 everywhere. What bound does the comparison property give for the integral from 0 to 1 of f?
9. Sum of squares formula
Direct evaluation of the integral of x^2 from 0 to 1 uses the sum of the first n squares. What is that formula?
10. Average value connection
The average value of a continuous function f on [a, b] is defined as 1/(b - a) times the integral. What does the mean value theorem for integrals add to this definition?
11. Integral of a constant## practice-fill Integral of a constant
The integral from a to b of a constant c equals c times the interval length. What is the integral from 1 to 4 of 5 dx? Answer with a number.
12. Degenerate integral
The integral from 2 to 2 of any integrable function f equals what number? Answer with a number.
13. Comparison direction
On [0, 2], f(x) = x^2 and g(x) = x satisfy f at most g. Which is larger, the integral of f or the integral of g? Answer with f or g.
14. Absolute bound
The triangle inequality in integral form says |integral of f| is at most the integral of |f|. As a three-letter abbreviation, what is this inequality commonly called? Answer in lowercase.