Help understanding Definition of Definite Integral

[stillofthenight]

Here is my thought process on The Definition of a Definite Integral. Please see if you see any flaws because I am getting confused of how a Limit equals an area. I think of Limit is a sole single number ( a point on a curve) as x approaches a value and cannot see the connection of a single number being approached, and ENTIRE area under a curve.

I get that a Riemann Sum R(f, P, C) is a sum of areas of rectangles on a Partition of size N with sample points C. Now I get that ||P|| is the largest of the sub intervals and that the definition states to shrink ||P|| to zero...okay so it approaches zero and it will approach a single point on a curve which I suppose is the Limit L. Well how does that one point equal an entire area? I would assume you add up all the areas after ||P|| approaches zero, but the definition is saying mathematically that the Limit ( a single point on a curve) equals an entire area on a sub interval which is confusing me.

Again I keep thinking how can this one point on a curve , the Limit L, equal an entire area on sub interval [a, b ]

 

[Bob Brown MSEE]

Keep in mind, as the individual areas approach zero area, you are summing a number of areas that approaches infinite.

 

[stillofthenight]

yes. so how can summing infinite areas equal a single y value limit?

My book formally states the following:

L= lim ||P|| --> 0 of R(f,P,C)

How can a Limit , one point on a curve and one value, equal the summation of infinite rectangles which are infinite values?

 

[Bob Brown MSEE]

Try this

Click Here

 

[HallsofIvy]

Here is my thought process on The Definition of a Definite Integral. Please see if you see any flaws because I am getting confused of how a Limit equals an area. I think of Limit is a sole single number ( a point on a curve) as x approaches a value and cannot see the connection of a single number being approached, and ENTIRE area under a curve.

Okay, the limit is " sole single number" (but NOT "a point on a curve- a point is NOT a number). Do you not realize that the "ENTIRE area under a curve" is also "a sole single number"? Perhaps you are confusing "area" (a "sole single number" indicating the size of a two dimensional region) with "region" (the set of points with given boudary).

I get that a Riemann Sum R(f, P, C) is a sum of areas of rectangles on a Partition of size N with sample points C. Now I get that ||P|| is the largest of the sub intervals and that the definition states to shrink ||P|| to zero...okay so it approaches zero and it will approach a single point on a curve which I suppose is the Limit L. Well how does that one point equal an entire area? I would assume you add up all the areas after ||P|| approaches zero, but the definition is saying mathematically that the Limit ( a single point on a curve) equals an entire area on a sub interval which is confusing me.

Again I keep thinking how can this one point on a curve , the Limit L, equal an entire area on sub interval [a, b ]