Help with Integration algebra!

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Howdy felles.

I have a question which I find rather confusing, and that is to prove that the area between two curves is given by [tex]A=\frac{n-1}{n+1} [/tex] where n is the power of X.


The equation is [tex]A=\int X^\frac{1}{n} dx - \int X^n dx[/tex]

Could someone plz show me how to get this to equal to [tex]A=\frac{n-1}{n+1} [/tex]

I had a very diffrent answer. Plz help

Cheers
 

Answers and Replies

Are there limits on those integrals ?
 
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The limits are from 0 to 1
 
lurflurf
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First do the integrals
[tex]=\int_0^1 x^\frac{1}{n} dx[/tex]
and
[tex]=\int_0^1 x^n dx[/tex]
Then simplify and get a common denominator.
What did you get for the integrals?
 
Last edited:
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I am preety sure mine was wrong as I ended up with a diffrent answer. Can some one plz check where I made mistake>

[tex]A=\int X^\frac {1}{n} dx - \int X^n dx[/tex]
[tex]A=\frac {[X^\frac {1}{n}+1]}{\frac {1}{n}+1} - \frac {[X^n+1}{n+1]}[/tex] Cant seem to make any progress from here onwards. All I do is make a massive mess of it and I have to do it again without any good signs.

Would really appriciate if anyone could help me.
 
202
0
I am preety sure mine was wrong as I ended up with a diffrent answer. Can some one plz check where I made mistake>

[tex]A=\int X^\frac {1}{n} dx - \int X^n dx[/tex]
[tex]A=\frac {X^\frac {1}{n+1}}{\frac {1}{n}+1} - \frac {X^n+1}{n+1}[/tex] Cant seem to make any progress from here onwards. All I do is make a massive mess of it and I have to do it again without any good signs.

Would really appriciate if anyone could help me.
 
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Also the limits of all these integrations are 0 to 1
 
VietDao29
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Okay, you should remember that:
[tex]\int^{\beta}_{\alpha} f(x)dx = F(\beta) - F(\alpha)[/tex]
So I'll do for you the first one. And you can follow it to do the second:
[tex]\int^{1}_{0} x ^ {\frac{1}{n}}dx = \left[ \frac{x ^ {\frac{1 + n}{n}}}{\frac{1 + n}{n}} \right] ^{1}_{0} = \frac{1 ^ {\frac{n + 1}{n}} - 0 ^ {\frac{n + 1}{n}}}{\frac{1 + n}{n}} = \frac{n}{1 + n}[/tex]
So can you do the second integral?
Viet Dao,
 
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I am still having abit trouble .

I am sposed to have [tex]A=\frac {n}{1+n} - \int^{o}_{1} X^n dx [/tex]

which I ended up getting [tex]\int^{1}_{0} X^n dx = \frac {1^n^+^1 - 0^n^+^1}{n+1}[/tex]
does it [tex]=\frac {n}{n+1}[/tex]
 
VietDao29
Homework Helper
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Nope. Your integral is 'nearly' correct. The last part is wrong.
Remember : [tex]1 ^ m = 1, m \in \mathbb{R}[/tex]
[tex]0 ^ m = 0, m \in \mathbb{R} - \{0\}[/tex]
So [tex]1 ^ {n + 1} = 1[/tex]
Viet Dao,
 
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