What does it mean to integrate a parameter vector? A naive question: I know a bout integrating a scalar function over values of x $$\int f(x)dx $$ and I'm trying to learn Machine learning now, however I face integrals integrating parameter vector w over dw $$\int f(W)dW $$ so what does this integral mean? For example, what is the riemann sum that this integral is calculating? Is this the same as line integral? 
An example below

 A: It's usually just a regular multidimensional integral.
Just think of parameter space as you would any other space.
(It won't be a more complicated line or surface integral unless the parameters form some kind of manifold of positive co-dimension because you have constrain relations between variables).
For instance, integrating over the parameters of a linear regressor $ g_{\alpha,\beta}(x)= \alpha x + \beta $ in 1D:
$$f(x) = \iint J(\alpha x + \beta) d\alpha\,d\beta $$

For Bayesian models, it can be a bit confusing, so let me give an example using them. Suppose we have a 1D regressor model $\widehat{y} = f_\theta(x)$ with parameters $\theta$. Given a dataset $ D=\{(x_i,y_i)\}_i$, we usually have a likelihood
like $$ p(y_i|x_i,\theta)=\mathcal{N}(y_i|f_\theta(x_i),\sigma^2_\ell) $$
as well as a prior over the weights $$ p(\theta) = \mathcal{N}(\theta|0,\sigma^2_pI) = \prod_d \mathcal{N}(\theta_d|0,\sigma^2_p) $$
Now, we need our model to give us a prediction on some new input ${x_\text{new}}$. We need the predictive distribution:
\begin{align} p(y_\text{new}|x_\text{new},D)
&= \int p(y_\text{new}|x_\text{new},D,\theta) p(\theta|x_\text{new},D) d\theta \\
&= \int p(y_\text{new}|x_\text{new},\theta) p(\theta|D) d\theta\end{align}
The first term is the likelihood, which is not a big deal, but the second is the posterior over the parameters:
$$ p(\theta|D)=\frac{p(D|\theta)p(\theta)}{p(D)} $$
which has the prior, the likelihood over the training set $$ p(D|\theta) = \prod_i p(y_i|x_i,\theta)  $$
and the model evidence (marginal likelihood)
$$ p(D) = \int p(D|\theta) p(\theta)d\theta $$
which can (sometimes) be ignored, since it does not depend on the parameters.
Check out this link too.
