黎曼和 mesh→0

hbghlyj

GR1268第11题
The region bounded by the curves $y=x$ and $y=x^2$ in the first quadrant of the $x y$-plane is rotated about the $y$-axis. The volume of the resulting solid of revolution is ___
答案：

We will find a formula in terms of $y_i$ for the volume of a thin washer $\Delta V_i$, and then formulate the total volume as the limit of a Riemann sum of $\Delta V_i$ 's. Each slice of volume is the area of its annular cross-section multiplied by a small vertical length $\Delta y_i$. The outer radius of the annular cross-section lies on the curve $y=x^2$ and inner radius on the line $y=x$. Hence, at the $y$-value $y_i$ the outer and inner radii are $\sqrt{y_i}$ and $y_i$, respectively. Thus, $$ \Delta V_i=\pi\left(\left(\sqrt{y_i}\right)^2-\left(y_i\right)^2\right) \Delta y_i=\pi\left(y_i-\left(y_i\right)^2\right) \Delta y_i $$

The minimum and maximum $y$-values, within the region bounded by $y=x$ and $y=x^2$, are 0 and 1 , respectively. Consider the partition $P=\{1,1 / 2,1 / 3, \ldots, 1 /(n+1)\}$ of the interval $[0,1]$. Suppose $y_i=1 /(n-i+1)$ and $\Delta y_i=1 /(n-i+1)-1 /(n-i+2)$. Notice that $\sum_{i=1}^n \Delta V_i \approx V$. Furthermore, as $n \rightarrow \infty$, we have $\sum_{i=1}^n \Delta V_i \rightarrow V$ and $$ \sum_{i=1}^n \Delta V_i=\sum_{i=1}^n \pi\left(y_i-\left(y_i\right)^2\right) \Delta y_i \longrightarrow \int_0^1 \pi\left(y-y^2\right) d y . $$ Hence, the volume obtained from rotating the region bounded by $y=x$ and $y=x^2$ about the $y$-axis is $$ V=\int_0^1 \pi\left(y-y^2\right) d y=\pi\left[\frac{y^2}{2}-\frac{y^3}{3}\right]_0^1=\frac{\pi}{6} . $$We select (B) and continue. Note that we went through the reasoning behind the washer method.

我的问题：$P=\{1,1 / 2,1 / 3, \ldots, 1 /(n+1)\}$是$[0,1]$的划分，但划分子区间长度最大值是$\frac12$不趋于0，所以它的黎曼和不趋于积分吧？

hbghlyj

Riemann integral

One important requirement is that the mesh of the partitions must become smaller and smaller, so that in the limit, it is zero. If this were not so, then we would not be getting a good approximation to the function on certain subintervals.

hbghlyj

$P$取错了，但它后面没用到这个$P$，答案是对的。