  Let $K$ be a field of characteristic zero, $R = K[X_1,...,X_n]$. Let $A_n(K)
= K<X_1,...,X_n, \partial_1, ..., \partial_n>$ be the $n^{th}$ Weyl algebra
over $K$. We consider the case when $R$ and $A_n(K)$ is graded by giving $\deg
X_i = \omega_i $ and $\deg \partial_i = -\omega_i$ for $i =1,...,n$ (here
$\omega_i$ are positive integers). Set $\omega = \sum_{k=1}^{n}\omega_k$. Let
$I$ be a graded ideal in $R$. By a result due to Lyubeznik the local cohomology
modules $H^i_I(R)$ are holonomic $A_n(K)$-modules for each $i \geq 0$. In this
article we prove that the De Rahm cohomology modules $H^*(\bP ; H^*_I(R))$ is
concentrated in degree $- \omega$, i.e., $H^*(\bP ; H^*_I(R))_j = 0$ for $j
\neq - \omega$. As an application when $A = R/(f)$ is an isolated singularity
we relate $H^{n-1}(\bP ; H^1_{(f)}(R)$ to $H^{n-1}(\partial(f); A)$, the
$(n-1)^{th}$ Koszul cohomology of $A$ \wrt \ $
\partial_1(f),...,\partial_n(f)$.
