Calculus Volume 2 · 免费预览
面积的近似计算
曲线下的面积可以用矩形面积之和来近似，分别采用左端点、右端点和中点取样，并通过不断细分使近似值逼近定积分所定义的精确值。
1. The area problem returns
The area problem returns
2. Left, right, and midpoint sums
Left, right, and midpoint sums
3. Refinement and convergence
Refinement and convergence
4. Summation notation and shortcuts
Summation notation and shortcuts
5. What a Riemann sum approximates
A Riemann sum for f on [a, b] with n rectangles is fundamentally an approximation of what quantity?
6. Midpoint sum advantage
For a monotone function on an interval, why does the midpoint rule typically beat both endpoint rules?
7. Left and right sums bracket
For an increasing function on [a, b], how do the left and right Riemann sums relate to the exact area?
8. Effect of doubling n
For the left and right endpoint rules, doubling the number of rectangles roughly does what to the error?
9. Delta x computation
The interval [2, 5] is partitioned into 6 equal subintervals. What is the width Delta x of each subinterval?
10. Sample point flexibility
The definition of the definite integral as a limit of Riemann sums requires the limit to be the same regardless of the sample points x_i star chosen in each subinterval. Why is this independence important?
11. Left sum for identity## practice-fill Left sum for identity
For f(x) = x on [0, 1] with n = 4 rectangles of width 1/4, the left endpoints are 0, 1/4, 1/2, 3/4. What is the left Riemann sum? Answer with a fraction like 3/8.
12. Rectangle area structure
Each term of a Riemann sum has the form f(x_i star) times Delta x. What does this term represent geometrically? Answer in two words like strip width.
13. Midpoint sample of a strip
A strip spans [x_{i-1}, x_i]. The midpoint rule samples the function at the strip's center. What is that center, expressed with x_{i-1} and x_i? Answer like (a + b)/2.
14. Exact area under x squared
The Riemann sums for x^2 on [0, 1] converge to a famous exact value as n grows. What is that value, as a fraction? Answer like 1/3.