Determine the area of a region between two curves defined by algebraic functions

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Discussion Overview

The discussion revolves around determining the area of a region bounded by two algebraic functions, specifically f(x) = 3√x - 4 and g(x) = 3x/5 - 8/5. Participants are exploring the mathematical steps necessary to find this area, including identifying the correct functions and their intersections.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • Post 1 presents the problem of finding the area A of the region R bounded by the two functions.
  • Post 2 reiterates the problem and asks what has been done so far regarding the solution.
  • Post 3 suggests finding the roots of the equation 3√x - 4 = 3x/5 - 8/5 to establish the endpoints for the integral needed to calculate the area.
  • Post 4 questions the definition of the first function, asking for clarification on whether it is f(x) = √x - 4 or f(x) = √(x - 4), and also challenges the interpretation of the second function g(x).
  • Post 5 proposes that f(x) = 3√x - 4 is the correct interpretation, as it leads to a simpler solution when equating it to g(x).
  • Post 6 echoes the question about the first function and reiterates the confusion regarding the second function, indicating a need for clarification.

Areas of Agreement / Disagreement

Participants express uncertainty regarding the definitions of the functions involved, with multiple interpretations presented. There is no consensus on the correct forms of f(x) and g(x), indicating a disagreement that remains unresolved.

Contextual Notes

Participants have not reached a clear agreement on the definitions of the functions, which may affect the calculation of the area. The discussion includes potential misinterpretations of the functions and their forms.

sgalos05
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R is the region bounded by the functions f(x)=3√x−4 and g(x)=3x/5−8/5. Find the area A of R. Enter answer using exact values.
 
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Beer soaked query follows.
sgalos05 said:
R is the region bounded by the functions f(x)=3√x−4 and g(x)=3x/5−8/5. Find the area A of R. Enter answer using exact values.
What have you done so far?
 
sgalos05 said:
R is the region bounded by the functions f(x)=3√x−4 and g(x)=3x/5−8/5. Find the area A of R. Enter answer using exact values.

You need the roots $a,b$ of the equation $3\sqrt{x}-4=3x\sqrt{5}-\frac{8}{5}$

Once you have established these roots use them as endpoints in

$\int_{a}^{b}\left(3\sqrt{x}-4-3x\sqrt{5}+\frac{8}{5}\right)dx$

The result is the area $A$ of $R$.
 
Is the first first function $f(x)= \sqrt{x}- 4$ or $f(x)= \sqrt{x- 4}$?

Also I do not see the second function as $g(x)= 3x\sqrt{5}- \frac{8}{5}$. I see $g(x)= \frac{3x}{5}- \frac{8}{5}= \frac{3x- 8}{5}$.
 
Just spitballin’ here ...

I would say $f(x) = 3\sqrt{x} - 4$ since it yields a simpler, nice solution when equating it to $g(x)$
 
Country Boy said:
Is the first first function $f(x)= \sqrt{x}- 4$ or $f(x)= \sqrt{x- 4}$?

Also I do not see the second function as $g(x)= 3x\sqrt{5}- \frac{8}{5}$. I see $g(x)= \frac{3x}{5}- \frac{8}{5}= \frac{3x- 8}{5}$.
yeah should have zoomed the page... lol... :)
 

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