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Re: Integration


Mark Dowell <mark.dowell at jrc.it>
[mg5865] Integration
writes

> I have a question involving the definite INTEGRATE function.
> With integration you give Mathematica; the function of x, and the  
values of
> x between which you want it integrated.  My problem is that I  
KNOW the area
> I want (and know that I am using the origin as xmin) but I need  
Mathematica
> to calculate xmax for me.

Mark:

You may find the following idea useful when your integral has to be  
solved numerically.	

Suppose we want to find for which X
	Integrate[Sqrt[Cosh[x] -1],{x,0,X}] = 4.

(!) Solve the corresponding differential equation to get a  
"Numerical Indefinite integral" over a suitable range:

NDSolve[{y'[x] == Sqrt[Cosh[x]-1], y[0]==0}, y, {x,0, 4}]
{{y -> InterpolatingFunction[{0., 4.}, <>]}}

yf = y/.%[[1]];

Plot[{yf[x],4},{x,0,4}]

	-Graphics-

FindRoot[yf[x] == 4, {x, {2,2.5}}]
	{x -> 3.05714}
Or

FindRoot[yf[x] == 4, {x, 2}, Jacobian -> Sqrt[Cosh[x]-1]]
	{x -> 3.05714}
	
NB. I got a value 3.071294 simply by clicking on the graph to  
select it then holding the Command key down and moving the pointer  
onto the intersection. The cordinates of the pointer are shown at  
the bottom of the window and if you click and copy they can be  
pasted.

Allan Hayes
hay at haystack.demon.co.uk
http://www.haystack.demon.co,uk


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