Geometric interpretation of \int x f'(x)

[Stephen Tashi]

I was reading Tom Apostol's expostion of Euler's Summation Formula ( http://www.jstor.org/pss/2589145) and it occurred to me that it would be convenient to visualize
$$\int_a^b x f'(x)$$
geometrically.

In that article, it arises from integration by parts:
$$\int_a^b f(x) dx = |_a^b x f(x) - \int_a^b x f'(x) dx$$

Visualize an increasing function f(x) in the first quadrant. The area given by $$|_a^b x f(x) = b f(b) - a f(a)$$ is the area of a big rectangle minus the are of a smaller rectangle. This are includes the area given by $\int_a^b f(x) dx.$

The area from the subtraction of the rectangular areas exceeds the area of that integral by an area that I can visualize as begin formed by horizontal line segments from the x-axis to the the curve of f(x) between the lines y= f(a) and y = f(b). In the integration by parts, the term $\int_a^b x f'(x) dx$ apparently subtracts that excess area in order to produce the right answer.

So, would it be good pedagogy to teach this as the geometric interpretation of $\int_a^b x f'(x) dx$ ? Is it in any textbook? I think it works out in the case when f(x) is decreasing also, since the sign of f'(x) makes the area negative.

 

[fresh_42]

I don't think that geometry is the best fit here. Integration by parts is essentially the Leibniz rule of differentiation, so it is more important to understand that.