39The formula for integration by parts for the definite integral has an interpretation based on the concept that the integral represents the area of the region enclosed by an axis and the graph of a positive function of the axis variable.
Thus ∫z1z2v(z)dz represents the area of the region enclosed by the z axis and the graph of v as a function of z while ∫v1v2z(v)dv represents the area of the region enclosed by the v axis and the graph of z as a function of v. From figure given below, it should seem reasonable that the sum of these two integrals is precisely the region with area v2z2-v1z1.
Or to summarize the result with definite integrals:
∫z1z2v(z)dz+∫v1v2z(v)dv=v2z2-v1z1.