<SYSTEM_TASK:>
Solve the following problem using Python, implementing the functions described below, one line at a time
<END_TASK>
<USER_TASK:>
Description:
 def _compute_integral_x(self):
 """Computes the integral of x, \\int_V x, over all atomic "triangles", i.e., areas cornered by a node, an edge midpoint, and a circumcenter. """

        # The integral of any linear function over a triangle is the average of
        # the values of the function in each of the three corners, times the
        # area of the triangle.
        right_triangle_vols = self.cell_partitions

        node_edges = self.idx_hierarchy

        corner = self.node_coords[node_edges]
        edge_midpoints = 0.5 * (corner[0] + corner[1])
        cc = self.cell_circumcenters

        average = (corner + edge_midpoints[None] + cc[None, None]) / 3.0

        contribs = right_triangle_vols[None, :, :, None] * average

        return node_edges, contribs