Definition of Indefinite Integral (a.k.a. Antiderivatives)

We say a function $F$ is an antiderivative of another function $f$ if $F’=f$. The indefinite integral $\int f(x)dx = F(x) + C\\ \\ \text{for some constant }C$ is the general form of antiderivatives of $f$.

Definition of Definite Integral

Riemann Sum

Give a partition $a=x_0<x_1<\cdots<x_n=b$ of $[a,b]$.
Choose sample points $x_i^*$ in each subinterval $[x_{i},x_{i+1}]$ for $i=0,1,\ldots, n-1$. Then, a $Riemann sum$ of $f$ on $[a,b]$ is $f(x_0^\*)(x_1-x_0)+f(x_1^\*)(x_2-x_1)+\cdots+f(x_{n-1}^\*)(x_{n}-x_{n-1})=\sum_{i=0}^{n-1}f(x_i^\*)(x_i-x_{i-1}).$

(See the following figure)

x = var('x') @interact def _(f=input_box("x^2+3", 'function'), a=input_box(1,type=float), b=input_box(3,type=float), number_of_partition=input_box(5, label='number of partition', type=int)): partition = [a] while len(partition)!=number_of_partition: x_i = RealField().random_element(a, b) if x_i not in partition: partition.append(x_i) partition.sort() partition.append(b) sample_points = [RealField().random_element(partition[i],partition[i+1]) for i in range(number_of_partition)] rectangles = [polygon2d([(partition[i], 0), (partition[i+1], 0), (partition[i+1], f.subs(x == sample_points[i])), (partition[i], f.subs(x == sample_points[i]))], alpha=0.2, edgecolor="black") for i in range(number_of_partition)] (plot(f,x,a,b)+point(partition)+sum(rectangles)).show(aspect_ratio="automatic", title="A Riemann sum of "+str(f)+" use "+str(number_of_partition)+" many rectangles") print('This Riemann sum is '+ str(sum([f.subs(x == sample_points[i])*(partition[i+1]-partition[i]) for i in range(number_of_partition)]))+',and the area under the curve from '+str(a)+' to '+str(b)+' is '+str(f.integral(x,a,b)))

考型 Give you a summation, and recognize it as a Riemann sum.

Example 1. Suppose the following summation is a Riemann sum of a function $f(x)$ on $[a,b]$ $\sum_{n=1}^m \frac{1}{\sqrt{n}(\sqrt{n+1})}$ Find $f$, $a$, and $b$.

Definition of Indefinite Integral

Definition of Definite Integral We say a function $f(x)$ is integrable on $[a,b]$ and has an integral $S$ if, for every $\varepsilon>0$, there exists a $\delta>0$ such that, for all partitions $a=x_0<x_1<\cdots<x_n=b$ with $\displaystyle\max_{i=1,2,\ldots, n}\{x_{i}-x_{i-1}\}<\delta$ and for arbitrary sample points $x_i^*\in[x_{i},x_{i-1}]$ with $i=1,2,\ldots,n$, we have $\left|S-\sum_{i=1}^nf(x_{i}^*)(x_i-x_{i-1})\right|<\varepsilon.$

We denote the integral of $f$ from $a$ to $b$ by $\int_{a}^b f(x)dx$.

考型 Use the definition of definite integral to find the integral of a function on $[a,b]$.

Example 2. Use the definition of definite integral to find the integral $\displaystyle\int_1^4 x^3+x^2dx$.

考型 Give you a limit of summation, and recognize it as an integral, and use this integral to find the limit.

Example 2. Evaluate the following limit $\displaystyle\lim_{m\to\infty}\sum_{n=1}^m\frac{1}{\sqrt{n}\sqrt{n+1}}.$