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Integrating over a disk

  • Thread starter Bre Ntt
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  • #1
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I'm taking a probability class where multivariate calculus was not a prerequisite, but some of it is coming up, I get the concept of, say integrating over a region, but get lost in some of the mechanics

Here is the problem (I don't want a full solution):


A point is uniformly distributed within the disk of radius 1. That is its density is
[itex]
f(x,y) = C \hspace{1cm} 0 \leq x^2 + y^2 \leq 1
[/itex]
Find the probability that its distance from the origin is less than x, 0 \leq x \leq 1

I'm pretty sure I have to set up an integral that integrates over a disc of radius x to get the probability
Something like this
[itex]
\int_A \int_B C \, dx \, dy
[/itex]
But I don't know what the intervals A and B are supposed to be.

Can someone point me in the right direction? I get confused because my attempts end up with x being involved in the limit of integration, but x is the dummy variable, which doesn't seem right.
 
Last edited:

Answers and Replies

  • #2
33,265
4,965
I'm taking a probability class where multivariate calculus was not a prerequisite, but some of it is coming up, I get the concept of, say integrating over a region, but get lost in some of the mechanics

Here is the problem (I don't want a full solution):


A point is uniformly distributed within the disk of radius 1. That is its density is
[itex]
f(x,y) = C \hspace{1cm} 0 \leq x^2 + 1 \leq 1
[/itex]
Shouldn't the inequality be 0 <= x2 + y2 <= 1?
Find the probability that its distance from the origin is less than x, 0 \leq x \leq 1
I'm pretty sure I have to set up an integral that integrates over a disc of radius x to get the probability
Something like this
[itex]
\int_A \int_B C \, dx \, dy
[/itex]
But I don't know what the intervals A and B are supposed to be.

Can someone point me in the right direction? I get confused because my attempts end up with x being involved in the limit of integration, but x is the dummy variable, which doesn't seem right.
 
  • #3
5
0
Yes, sorry. 0 <= x^2 + y^2 <= 1

I fixed it above.
 
  • #4
33,265
4,965
This would be a natural for polar form of a double integral.
 

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