1. ## Evaluating definite integral

I know the value of the following definite integral

$\displaystyle \int_{a}^{b}ydx$

I also have a realtion

$\displaystyle x=f(y)$

i.e. x is an explicit function of y but I do not have y as an explicit
function of x. The relation between x and y is generally non linear.

Now I want to get the following definite integral

$\displaystyle \int_{a}^{b}\left[\int ydx\right]xdx$

i.e. $\displaystyle \int ydx$ multiplied by x evaluated over the interval [a,b].

Is there an analytic (not numeric) way to evaluate this integral using
for example mean value or similar averaging technique?

2. ## Re: Evaluating definite integral

Can you actually state the problem you have instead of using generalities?

3. ## Re: Evaluating definite integral

There is no generality apart from $\displaystyle x=f(y)$.
So assume
$\displaystyle x=1+y-3y^3-9sin(y)$

4. ## Re: Evaluating definite integral

To find $\displaystyle \int y dx$ you can first get the derivative of f(y)

$\displaystyle \frac{dx}{dy}=f'(y)$

Rearrange for dx

$\displaystyle dx=dy f'(y)$

Then change the variable in the integral

$\displaystyle \int y (f'(y) dy)$

After that do integration by parts. By letting dv= f'(y) dy, you can see simply that v=f(y)

After the integration by parts suppose that the answer you get is g(y)

Then to find

$\displaystyle \int_a^b g(y) x dx$

Do the same trick as before to change the dx to dy, and substitute f(y) for x. And also change the limits of the integral to the new variable. So you get

$\displaystyle \int_{f^{-1}(a)}^{f^{-1}(b)} g(y) f(y) f'(y) dy$

Since you dont have y as an explicit function of x you will have to find $\displaystyle f^{-1}(a)$ by numerical methods

5. ## Re: Evaluating definite integral

Originally Posted by JulieK
I know the value of the following definite integral

$\displaystyle \int_{a}^{b}ydx$

I also have a realtion

$\displaystyle x=f(y)$

i.e. x is an explicit function of y but I do not have y as an explicit
function of x. The relation between x and y is generally non linear.

Now I want to get the following definite integral

$\displaystyle \int_{a}^{b}\left[\int ydx\right]xdx$
The way you have written this makes no sense. You need different variables for the two integrals. One way to write it would be $\displaystyle \in_a^b\int_0^x tf^{-1}(t)dt dx$.
If f is not invertible that integral is not defined.

i.e. $\displaystyle \int ydx$ multiplied by x evaluated over the interval [a,b].

Is there an analytic (not numeric) way to evaluate this integral using
for example mean value or similar averaging technique?