Area under a function befuddlement

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Discussion Overview

The discussion revolves around the conceptual understanding of integrating a function, specifically focusing on the area under a linear function with a slope of 3/4. Participants explore the implications of definite integrals over varying limits and the visualization of areas represented by trapezoids and rectangles.

Discussion Character

  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant describes a scenario where integrating a function with a slope of 3/4 leads to consistent area results regardless of the limits chosen, raising a conceptual confusion about the visualization of area as the limits increase.
  • Another participant questions the initial claim by pointing out a lack of clarity regarding the function being integrated, suggesting that the constant function f(x) = 3/4 was not correctly identified.
  • A different participant distinguishes between rectangles and trapezoids in the context of integration, indicating that the area calculations differ based on the shape and slope of the function being integrated.
  • One participant acknowledges the confusion and clarifies that they were referring to the area under a trapezoid, expressing difficulty in visualizing how areas remain equivalent despite increasing heights at larger x-values.
  • Another participant reiterates the distinction between the areas of rectangles and trapezoids, emphasizing that while the width remains constant, the heights vary, affecting the area of trapezoids.
  • A later reply reflects a moment of realization about the misunderstanding of integrating the function versus its derivative, indicating a shift in the participant's perspective.

Areas of Agreement / Disagreement

Participants express differing views on the nature of the function being integrated and the implications of the area calculations. There is no consensus on the conceptual visualization of the areas as limits increase, indicating ongoing confusion and debate.

Contextual Notes

Participants have not fully resolved the assumptions regarding the function being integrated, nor have they clarified the implications of the varying heights of trapezoids in relation to their areas. The discussion remains open to interpretation and further exploration.

DiracPool
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I'm having a conceptual problem with integrating a function and thought someone might enlighten me.

Say you have a simple function which is the simple slope 3/4. You take the definite integral of it an come up with 3/4 x evaluated between two values of x. Good so far.

Now, if I set the limits of integration between 0 and 4, I get a value of 3. We'll call the area, then, 3 meters squared. Good so far again.

Going further, if I set the limits of integration for high values of x with the same difference of 4 units, I still get the value of three when I plug in the limits of integration. For example, setting the limits at 400 to 396, still gives an answer of 3 square meters.

My conceptual problem is that, when I visualize this slope over 400 units of x, I see the same horizonal displacement on the graph, but I see an ever increasing vertical displacement. This leads me to believe that the area of each rectangular "slice" must be growing for higher values of x. But the maths say they are the same. I'm befuddled.

What is going on here, are the "slices" of 4 units getting thinner and thinner for higher values of x or am I missing something here?
 
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DiracPool said:
Say you have a simple function which is the simple slope 3/4.

You failed to state what function you are talking about.

The answers you proposed are not correct for integrating the function f(x) = (3/4)x.

Perhaps you are integrating the constant function f(x) = 3/4. That function has zero slope.
 
Is your shape a rectangle or a trapezoid? You may have confused the two.

[tex]\int_a^b \frac{3}{4}dx=\frac{3}{4}(b-a)[/tex]

[tex]\int_a^b \frac{3}{4}x dx=\frac{3}{8}(b^2-a^2)[/tex]
 
Sorry for the confusion gang, I am referring to the first equation put up by Lurflurf, with the solution 3/4 (b-a). And, yes, I guess it would be a trapezoid with that constant slope.

Again, though, I don't have a problem with the mathematics, I've just having a problem visualizing how the area of an extremely tall trapezoid when b=400 and a=396 is equivelant to the area of a trapezoid with b=4 and a=0. It's a conceptual problem. Maybe there isn't an answer...It just doesn't seem like the trapezoids are getting any thinner as you move to higher values of x.
 
The first one is a rectangle, its slope is zero. The second one is a trapezoid, its slope is 3/4. For given (b-a) like 4 in your example the area of the rectangle is the same, the trapezoids of width 4 have different areas due to their variation in height.
 
The first one is a rectangle, its slope is zero. The second one is a trapezoid, its slope is 3/4. For given (b-a) like 4 in your example the area of the rectangle is the same, the trapezoids of width 4 have different areas due to their variation in height.

Ok, thanks for the help guys. I think I was thinking of integrating the function rather than its derivative. Oy veh! I must be coming down with a seasonal cold or something:confused:
 

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