Finding the Area of a Triangular Region in the First Quadrant

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Discussion Overview

The discussion revolves around finding the area of a triangular region in the first quadrant, specifically bounded above by the curve y=e^(2x), below by y=e^x, and on the right by the line x=ln(3). The scope includes mathematical reasoning and integration techniques.

Discussion Character

  • Mathematical reasoning
  • Homework-related
  • Technical explanation

Main Points Raised

  • One participant expresses confusion about how to start the problem of finding the area.
  • Another suggests plotting the curves and checking definite integrals as a method to approach the problem.
  • A participant asks for clarification on how to check for definite integrals, noting that their textbook is not very descriptive.
  • It is proposed that the area can be found by determining integration limits after plotting the area.
  • One participant emphasizes the importance of understanding area given by integrals in solving such problems.
  • Another participant mentions that the curves intersect at x=0 and are bounded on the right by x=ln(3), suggesting the area can be expressed as an integral from 0 to ln(3) of (e^(3x) - e^x)dx.
  • There is a suggestion to estimate the area, questioning if it can be shown to be less than 1,000 or even 10.
  • A later reply reiterates the intersection points and provides specific values for the curves at x=ln(3), suggesting the area can be calculated using definite integrals of the respective functions.
  • Another participant outlines the steps to find the area, including integrating e^x and e^(2x) with specified limits.

Areas of Agreement / Disagreement

Participants generally agree on the need to use definite integrals to find the area, but there are varying degrees of understanding and clarity regarding the steps involved and the specific calculations. The discussion remains unresolved regarding the exact method and calculations to be used.

Contextual Notes

There are limitations in the discussion regarding the clarity of integration techniques and the specific steps required to solve the problem. Some participants express uncertainty about the integration process and the interpretation of the area under the curves.

shadow5449
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This one has me stumped, I don't even know where to start.
-Find the area of the triangular region in the first quadrant that is bounded above by the curve y=e^(2x), below by y=e^x, and on the right by the line x=ln(3).
Thanks for any help
 
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Try plotting the curves, and check definite integrals.
 
How would I go about checking for definite integrals? The book we use is not very descriptive on this topic. Thanks.
 
Plot the area, then find your integration limits. That's what he means by definite integral. Your final result will be a number, not an expression, representing the area in the shape.
 
If you are doing a problem like this, then you certainly should be familiar with area given by integrals- that's the whole point of a problem like this. If you graph this, you will see that the two curves intersect at x= 0 and are bounded on the right by the vertical line x= 3. If you divide the area into thin rectangles, the height of each rectangle will be e3x- ex and the width will be "dx". Now, what integral is that?
 
have you tried estimating it? i.e. can you show the area is less than 1,000?

or 10?
 
Since this has been around for a couple of days now:

The graph of y= e3x is always above y= ex for x> 0. The two graphs cross at x= 0 and we are told that the are is bounded on the right by x= ln 3.

The area is [tex]\int_0^{ln 3}(e^{3x}- e^x)dx[/tex].

Can you do that integral?
 
first draw the graph of all three fuctions. you'll get the picture. find the point of intersection of e^x, e^2x and x = ln3. this can be done easily...

one point is (0,1)...trivial solution
put x=ln3 in e^x and get y=3...(ln3,3)
put x=ln3 in e^2x and get y=9...(ln3,9)

now, for the next part you have to draw the graph.

integrate e^x limits 0 to 3 to get e^3 - 1
integrate e^2x limits 0 to 9 to get (e^9 - 1)/2

The area required is [(e^9 - 1)/2 - (e^3 - 1)]
note: area under graph of a function between a and b is its definite integral with a and b as lower and upper limits.
 

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