R help - May 2012 - Integration of two dimension function

Dear all,
   I have a function f <- function(x,y,c){as.numeric(x*y < c) }
  I need to solve the value of c so that when I take the integral of
the function f from 0.2 to 0.8 with respect to x and from 0 to 1
with respect to y, the integral equal some prefixed value, say 0.025.
  It involves two dimension integration. Can anyone give some
help on this?
  Thank you so much in advance.
       Hannah

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On Thu, May 17, 2012 at 09:43:13PM -0400, li li wrote:
> Dear all,
>    I have a function f <- function(x,y,c){as.numeric(x*y < c) }
>   I need to solve the value of c so that when I take the integral of
> the function f from 0.2 to 0.8 with respect to x and from 0 to 1
> with respect to y, the integral equal some prefixed value, say 0.025.
Hi.

The function f() is a 0/1 function. So, the integral may be 
reformulated as the area of the set of points (x, y) satisfying

  0.2 <= x <= 0.8
  0   <= y <= 1
  x*y <= c

In other words, it is the univariate integral of the function

  g(x) = min(1, c/x)

over the range x \in [0.2, 0.8]. This is 
     
  G <- function(cc) {
    ifelse(cc <= 0.2, cc*log(4), # cc*(log(0.8) - log(0.2))
           ifelse(0.2 < cc & cc < 0.8, (cc - 0.2) + cc*(log(0.8) -
log(cc)), 0.8 - 0.2))
  }

What i get is that G(cc) = 0.025 for cc = 0.025/log(4).

I hope there are not too may errors in this calculation.

Petr Savicky.