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

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SUMMARY

The area of the region R bounded by the functions f(x) = 3√x - 4 and g(x) = 3x/5 - 8/5 can be calculated using definite integrals. The roots of the equation 3√x - 4 = 3x/5 - 8/5 must be determined to establish the limits of integration. The area A of R is found using the integral A = ∫ab(3√x - 4 - (3x/5 - 8/5))dx, where a and b are the roots of the equation. The correct interpretation of the functions is critical for accurate calculations.

PREREQUISITES
  • Understanding of definite integrals
  • Knowledge of algebraic functions and their intersections
  • Ability to solve equations involving square roots
  • Familiarity with the Fundamental Theorem of Calculus
NEXT STEPS
  • Calculate the roots of the equation 3√x - 4 = 3x/5 - 8/5
  • Perform the definite integral ∫ab(3√x - 4 - (3x/5 - 8/5))dx
  • Explore the properties of definite integrals in calculus
  • Review the application of the Fundamental Theorem of Calculus
USEFUL FOR

Students and educators in calculus, mathematicians focusing on area calculations between curves, and anyone interested in applying integral calculus to solve real-world problems involving algebraic functions.

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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