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

In summary: So in summary, R is the region bounded by the functions f(x)=3√x−4 and g(x)=3x/5−8/5. To find the area A of R, you need to find the roots $a,b$ of the equation $3\sqrt{x}-4=3x\sqrt{5}-\frac{8}{5}$ and use them as endpoints in $\int_{a}^{b}\left(3\sqrt{x}-4-3x\sqrt{5}+\frac{8}{5}\right)dx$. This will give you the area A of R. There was a question about the first function being $f(x)= \sqrt{x}- 4$ or
  • #1
sgalos05
1
0
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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  • #2
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?
 
  • #3
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 $R$.
 
  • #4
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 $g(x)= \frac{3x}{5}- \frac{8}{5}= \frac{3x- 8}{5}$.
 
  • #5
Just spitballin’ here ...

I would say $f(x) = 3\sqrt{x} - 4$ since it yields a simpler, nice solution when equating it to $g(x)$
 
  • #6
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... :)
 

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

How do I determine the area between two curves using algebraic functions?

The area between two curves can be determined by finding the definite integral of the function that represents the upper curve minus the definite integral of the function that represents the lower curve. This will give you the area between the two curves within the specified boundaries.

What are the steps involved in finding the area between two curves?

The steps involved in finding the area between two curves are:
1. Identify the two curves and determine which one is the upper curve and which one is the lower curve.
2. Set up the definite integral by subtracting the lower curve's definite integral from the upper curve's definite integral.
3. Determine the boundaries of the region between the two curves.
4. Evaluate the definite integral to find the area between the two curves within the specified boundaries.

Can I use any algebraic functions to find the area between two curves?

Yes, you can use any algebraic functions as long as they represent the upper and lower curves. The functions must also be continuous within the specified boundaries.

How do I determine the boundaries of the region between two curves?

The boundaries of the region between two curves can be determined by finding the points of intersection between the two curves. These points will serve as the upper and lower limits of integration for the definite integral.

Is there a specific formula for finding the area between two curves?

Yes, the formula for finding the area between two curves is:
A = ∫[a,b] (f(x) - g(x)) dx
where f(x) is the upper curve, g(x) is the lower curve, and [a,b] are the boundaries of the region between the two curves.

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