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## Main Question or Discussion Point

I am a beginner on integration and I am stuck on some concepts. I hope that someone can explain, thank you so much!

1) "Let f be continuous on [a,b].

Take a partition P={xo,x1,...,xn} of [a,b].

Then P breaks up [a,b] into n subintervals [xo,x1],[x1,x2],...[x(n-1),xn] of lengths delta x1, delta x2,... delta xn.

Now pick a point x1* from [xo,x1] and form the product f(x1*)delta x1; pick a point x2* from [x1,x2], and form the product f(x2*)delta x2 and so on, we define the

We define ||P|| as the

(int_a^b f(x)dx is read the definite integral of f from a to b)

Then

lim S*(P) = int_a^b f(x)dx "

||P||->0

I don't understand why the limit of Riemann sum as ||P||->0 will be equal to the definite integral......

Say if I have delta x1,delta x2,...delta xn, with delta x1 being the largest, now if I only take ||P||->0, i.e. delta x1 ->0, all the other delta x2,...delta xn won't will be affected, then how is it possible that this is the definite integral?

Graphically, Riemann sum can be pictured as areas of rectangles under the curve, and now I only make the WIDEST rectangle narrower (||P||->0) while keeping all the other rectangles the same widths, now clearly the area of all rectangles & the area under the curve are unequal, then how can it be the definite integral?

2) Definite integral from a to b is int_a^b f(x)dx

What is the purpose of writing the "dx" at the end of the integral? Is it possible to have something like int_a^b f(x)dy, or int_a^b f(x)dz?

3) "A function f is integrable on an interval [a,b] if:

1. f is increasing on [a,b], or f is decreasing on [a,b]

2. f is bounded on [a,b] and has at most a finite number of discontinuites."

There is no key word in my textbook like "and", "or" between 1. and 2., so I am not sure whether both conditions need to be satisfied at the same time or is either one satisfied will be ok??

1) "Let f be continuous on [a,b].

Take a partition P={xo,x1,...,xn} of [a,b].

Then P breaks up [a,b] into n subintervals [xo,x1],[x1,x2],...[x(n-1),xn] of lengths delta x1, delta x2,... delta xn.

Now pick a point x1* from [xo,x1] and form the product f(x1*)delta x1; pick a point x2* from [x1,x2], and form the product f(x2*)delta x2 and so on, we define the

**Riemann sum**as S*(P)=f(x1*)delta x1 + f(x2*)delta x2 + ... + f(xn*)delta xnWe define ||P|| as the

**norm**of P, by setting ||P||=max delta xi, where i=1,2,...,n.(int_a^b f(x)dx is read the definite integral of f from a to b)

Then

lim S*(P) = int_a^b f(x)dx "

||P||->0

I don't understand why the limit of Riemann sum as ||P||->0 will be equal to the definite integral......

Say if I have delta x1,delta x2,...delta xn, with delta x1 being the largest, now if I only take ||P||->0, i.e. delta x1 ->0, all the other delta x2,...delta xn won't will be affected, then how is it possible that this is the definite integral?

Graphically, Riemann sum can be pictured as areas of rectangles under the curve, and now I only make the WIDEST rectangle narrower (||P||->0) while keeping all the other rectangles the same widths, now clearly the area of all rectangles & the area under the curve are unequal, then how can it be the definite integral?

2) Definite integral from a to b is int_a^b f(x)dx

What is the purpose of writing the "dx" at the end of the integral? Is it possible to have something like int_a^b f(x)dy, or int_a^b f(x)dz?

3) "A function f is integrable on an interval [a,b] if:

1. f is increasing on [a,b], or f is decreasing on [a,b]

2. f is bounded on [a,b] and has at most a finite number of discontinuites."

There is no key word in my textbook like "and", "or" between 1. and 2., so I am not sure whether both conditions need to be satisfied at the same time or is either one satisfied will be ok??

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