Integration by First Principles (1 Viewer)

[QZP]

From Cambridge, "To find a definite integral by first principles, dissect the interval into n equal subintervals, construct upper and lower rectangles on each subinterval, and find the sums of the upper and lower rectangles. Then their common limit will be the value of the integral."

My question is would it not suffice to use just either the upper OR lower rectangles and not both? I don't see the need for both :S

 

[fizzbylightning]

Maybe they want you to show that both upper and lower sums tend toward the same area and n tends to infinity. The area underneath the curve is between the lower and upper sum.

 

[Trebla]

The idea is that

Sum of areas of lower rectangles < Exact area < Sum of areas of upper rectangles

If the sum of the areas of both lower and upper rectangles converge to the same value as the number of rectangles approaches infinity then you can conclude that the exact area is that value. If you only show that the sum of areas of say lower rectangles only converges to a value then you can't really say anything about the exact area with much rigour.

 

[QZP]

The idea is that

Sum of areas of lower rectangles < Exact area < Sum of areas of upper rectangles

If the sum of the areas of both lower and upper rectangles converge to the same value as the number of rectangles approaches infinity then you can conclude that the exact area is that value. If you only show that the sum of areas of say lower rectangles only converges to a value then you can't really say anything about the exact area with much rigour.

Perfect answer, thanks

 

[anomalousdecay]

The idea is that

Sum of areas of lower rectangles < Exact area < Sum of areas of upper rectangles

If the sum of the areas of both lower and upper rectangles converge to the same value as the number of rectangles approaches infinity then you can conclude that the exact area is that value. If you only show that the sum of areas of say lower rectangles only converges to a value then you can't really say anything about the exact area with much rigour.

May I just add in here that this is a super important property in ext.2, which you are expected to use in some ext.2 exam questions.

 

[nightweaver066]

May I just add in here that this is a super important property in ext.2, which you are expected to use in some ext.2 exam questions.

Squeeze theorem that is.

 

[seanieg89]

From Cambridge, "To find a definite integral by first principles, dissect the interval into n equal subintervals, construct upper and lower rectangles on each subinterval, and find the sums of the upper and lower rectangles. Then their common limit will be the value of the integral."

My question is would it not suffice to use just either the upper OR lower rectangles and not both? I don't see the need for both :S

Yep, the problem is when you "visualise" that upper rectangles would suffice, I assume you have a nice and continuous function f in mind. If f is not continuous, there is no reason to be sure that upper Riemann sums and lower Riemann sums converge to the same fixed value (which we then call the integral of f).

If a upper and lower Riemann sums of a function DO converge to the same value, we say that the function is Riemann-integrable, and it makes sense to integrate it. Otherwise our function is "too wild" for us to define a sensible notion of integration (at least by using Riemann sums).

 

[anomalousdecay]

Squeeze theorem that is.

Is that the name of the questions with the hyperbolic functions that has come up in most of the exams?

 

[nightweaver066]

Is that the name of the questions with the hyperbolic functions that has come up in most of the exams?

It's a theorem, not a style of question.

Not sure which questions you mean though, have a read:
http://en.wikipedia.org/wiki/Squeeze_theorem

 

[fizzbylightning]

Aka pinching theorem