Calculus 2: Week 1

The first week of classes at my college went by quite quickly. My math class this quarter is Calculus 2. I am taking an evening class, which meets twice a week. This is quite a change from last quarter, when I had Calc 1 each day Monday through Friday.

In class, we started with the concept of an integral, defined as the Riemann sum of areas.

The Definite Integral

Suppose $f$ is defined on $[a,b]$. Let $[a,b]$ be divided into $n$ equal subintervals of width $\Delta x = (b-a)/n$. Now let $x_{0} = a$, $x_{1}$, $x_{2}$, $x_{i} = a + i\Delta x$, ... $x_{n} = b$ be the endpoints of the subintervals, and let $x_{i}*$ be any sample point in the $i$th interval $[x_{i-1}, x_{i}]$. Then

$$\int_{a}^{b} f(x) dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_{i}*)\Delta x$$

The precise meaning of this limit:

$\forall$ $\epsilon >0$ $\exists$ an integer $N$ such that

$$\left|\int_{a}^{b} f(x) dx - \sum_{i=1}^{n} f(x_{i}*)\Delta x \right| < \epsilon$$

 $\forall$ integers $n>N$ and $\forall$ $x_{i}* \in [x_{i-1}, x_{i}]$.


Next, we went over the Fundamental Theorem of Calculus, which shows that differentiation and integration are inverse processes.

Fundamental Theorem of Calculus

Suppose $f$ is continuous on $[a,b]$ and $F' = f$. Then

  $$\frac{d}{dx}\int_{a}^{x} f(t) dt = f(x)$$

$$\int_{a}^{b} f(x) dx = F(b) - F(a)$$


Wow, this took longer than I thought with all the Latex. I will try to make a post like this every week, if I am not too busy. Our class is using Stewart's Calculus (8th Edition), so my formulas, definitions, theorems, etc. for this and other posts are and will be pretty much copied from the textbook.

Coming up soon: indefinite integrals and u-substitution (reverse chain rule).