What is n for the shaded region?

[Emethyst]

Homework Statement

Find the value of n so that the area of the shaded region (refer to attached picture) in the following diagram is a) 50% of the area of the unit square b) 80% of the area of the unit square

Homework Equations

Definate integral properities, fundamental theorem of calculus

The Attempt at a Solution

My teacher says this is an easy question, but I cannot seem to solve it. My first guess was to take the integral of the two functions and then use the percentages as the answers for the integral, solving for n in each case (in this case I made 50% = 1/2 and 80% = 4/5). This did not get me very far for I came up with square roots and fractions in my answers when they should be simply be one number answers. Am I doing this question right through this method, or is this the wrong procedure and do I need to do something different to find the correct answers? Any help would be very welcome, thanks in advance. PS: sorry for the bad picture :tongue:

 

[Mark44]

The picture has been pending approval for several hours. Can you describe the region in words?

 

[Emethyst]

There are two functions, y = x^n and y = x^1/n located in region I of the graph. The area to be found is the area in-between these two functions, and runs from their point of intersection at (0,0) to the other point at (1,1). y = x^n is located below y = x^1/n (though I'm sure you probably already figured that out :tongue:).

 

[squidsoft]

$$A=\int_0^1\int_{x^n}^{x^{1/n}} dydx$$
(depending if n>1 or n<1)

Evaluate the integral then solve for n. I get:

$$\frac{n}{n+1}-\frac{1}{n+1}=A$$

 

[Emethyst]

I understand the last part, but I don't know how to get (n-1)/(n+1) from that integral you used. Unless you mean finding the integral of x^1/n - x^n over [0,1]?

 

[squidsoft]

Hi. Didn't notice you replied until now.

<< solution deleted by berkeman >>

 

[Emethyst]

Thanks squidsoft, I managed to solve that one.