Differences

This shows you the differences between two versions of the page.

--- math104-s21:hw11 [2021/04/16 22:28]
pzhou
+++ math104-s21:hw11 [2022/01/11 10:57] (current)
pzhou ↷ Page moved from math104-2021sp:hw11 to math104-s21:hw11
@@ Line 17: / Line 17: @@
 . (Free discussion problem, no points taken). Let $f: [0,1] \to \R$ be a real bounded function. Assume $f(x) \geq 0$ and $f$ is Riemann integrable. Here is a candidate that measures the area under the graph of $f$:
-  * For each positive integer $n$, let $N(n)$ be the number of points $(x,y) \in \R^2$, such that $x, y \in \Z / 2^n$, and $x \in [0,1], 0 \leq y \leq f(x)$. Namely, we count how many points in the lattice $2^{-n} \Z^2 \In \R^2$ falls within the area under the curve $y=f(x)$.
+  * For each positive integer $n$, let $N(n)$ be the number of points $(x,y) \in \R^2$, such that $x, y \in \Z / 2^n$, and $x \in [0,1], 0 \leq y \leq f(x)$. Namely, we count how many points in the 2d mesh with spacing $2^{-n}$ falls within the area under the curve $y=f(x)$.
   * Let $A(n) = 2^{-2n} N(n)$. Since the area of a $2^{-n} \times 2^{-n}$ square is $2^{-2n}$.
-Is it true that $\lim_{n \to \infty} A(n) = \int_0^1 f(x)dx$?
+Is it true that
+$$ \lim_{n \to \infty} A(n) = \int_0^1 f(x)dx ? $$
+To get started, you may assume that $f$ is continuous (we will see next week that continuous functions are Riemann integrable).

Lecture Notes