About fundamental theorem of calculus

[might]

You know;

if F(x) = INT f(x) dx

b
INT f(x) dx equals to F(b) - F(a)
a

But if INT f(x) dx can not be written in terms of elementary functions, and if I know the value of (a) and also the result of the integration, is there a way to calculate the value of (b)?

Thanks in advance.

[A$$Bandit]

You'd have to do it numerically using something like Simpson's Rule.

[might]

Thanks for your reply.

I know about Simpson`s 1/3 rule. Would you please share your idea to calculate this using it.

[krtxmrtz]

Call A = Integral from a to b of Int[f(x)]



A is the area bounded by the horizontal axis, the horizontal red lines and the vertical green lines.
The numbers i, i+1, i+2, ..., i+k-2, i+k-1, i+k are integers and I define

g_n = f^-1(n)

as the value of the inverse of f at those points.

Then the integral (area) can be written as:

A = i(g_i+1 - a) + (i + 1)(g_i+2 - g_i+1) + (i + 2)(g_i+3 -

g_i+2) + ... + (i + k - 2)(g_i+k-1 - g_i+k-2) + (i + k - 1)(b -

g_i+k-1) = i(b - a) + (g_i+2 - g_i+1) + 2(g_i+3 -

g_i+2) + 3(g_i+4 - g_i+3) + ... + (k - 2)(g_i+k-1 -

g_i+k-2) + (k - 1)(b - g_i+k-1) = i(b - a) - g_i+1 - g_i+2 -

g_i+3 - ... - g_i+k-2 - g_i+k-1 + (k - 1)b

Now notice that

i = INT[f(a)]

and

i + k - 1 = INT[f(b)]

so that:

A = (b - a)INT[f(a)] + b{INT[f(b)] - INT[f(a)]} - SUM(from j=1 to INT[f(b)] - INT[f(a) of (INT(f(a) +

j) = b INT[f(b)] - a INT[f(a)] - SUM(from j=1 to INT[f(b)] - INT[f(a) of (INT(f(a) + j)

(sorry, I couldn't think of any nice-looking way to write the sum over the index j)
so if you know A and a then you know b right away.

[krtxmrtz]

This is not so ugly-looking.

[krtxmrtz]

By the way, I've just noticed I was completely misled when I first read your post. So you just wanted the integral from a to b of f(x), not of INT(f(x)) !!! I'm glad I got it wrong for I had tons of fun working on it!

Well, as for your question, I think you shouyld go to places like the "Numerical Recipes" web site. There's a chapter devoted to numerical integration with lots of Simpson formulas and its variations and a lot more stuff. You can download the full book in pdf format. They have some theory and then the "recipe" in a number of computer languages. Unfortunately not in vb, but it shouldn't be difficult to translate from Fortran or c. For example, the address below corresponds to the c edition:

http://www.library.cornell.edu/nr/bookcpdf.html
(look at chapter 4)

If you want the main pages, go to Numerical Recipes

[might]

Thank you very much for your detailed post. But I only mean integral with "INT".

I only wanted to canculate (b).

if f(x) sqrt( (-a * sin(x))^2 + (b * cos(x))^2 )
b
I = integral f(x) dx
a

The result of the integration (I) and the value of (a) are known
and I`ve been trying to find a solution to get the value of (b).

The web site you linked is a very useful one as a solution and algorithm source. A type of a website I have been searching for.

Thanks, again.

[A$$Bandit]

To be honest I can't be arsed with calculating anything. Oh dear.