Does the centrifugal force really act on a rotating body?

No, centifugal force is an imaginary force.there will act no net force on a rotating body in outside radial direction.there is only one net force ,acting inwards radial,which depends on the speed of rotation and radius of curvature of path.

Why do we perform dynamic analysis for a mechanism?

These analyses are necessary to find out the forces acting on different members of the mechanism and there by find out stresses developed,so that, material of correct strength is chosen for proper running of the mechanism.

What is the geometric meaning of "derivative of a function at a point" and "integral of a function over an interval".

"derivative of a function at a point" means the slope of the tangent to the graph of the function at that particular point. "integral of a function over an interval" means the area in between the function curve and the x-axis 'in the given interval of x '.