Integration By Parts - Visualisation

Visualisation

Define a parametric curve by . Assuming that the curve is locally one-to-one, we can define

The area of the blue region is

Similarly, the area of the red region is

The total area is equal to the area of the bigger rectangle, minus the area of the smaller one, :

Assuming the curve is smooth within a neighborhood, this generalizes to indefinite integrals:

Rearranging:

Thus integration by parts may be thought of as deriving the area of the blue region from the total area and that of the red region.

This visualisation also explains why integration by parts may help find the integral of an inverse function when the integral of the function is known. Indeed, the functions and are inverses, and the integral may be calculated as above from knowing the integral .