- #1
- 7,861
- 1,598
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
[tex] \int_a^b x f'(x) [/tex]
geometrically.
In that article, it arises from integration by parts:
[tex] \int_a^b f(x) dx = |_a^b x f(x) - \int_a^b x f'(x) dx [/tex]
Visualize an increasing function f(x) in the first quadrant. The area given by [tex] |_a^b x f(x) = b f(b) - a f(a) [/tex] is the area of a big rectangle minus the are of a smaller rectangle. This are includes the area given by [itex] \int_a^b f(x) dx. [/itex]
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 [itex] \int_a^b x f'(x) dx [/itex] 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 [itex] \int_a^b x f'(x) dx [/itex] ? 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.
[tex] \int_a^b x f'(x) [/tex]
geometrically.
In that article, it arises from integration by parts:
[tex] \int_a^b f(x) dx = |_a^b x f(x) - \int_a^b x f'(x) dx [/tex]
Visualize an increasing function f(x) in the first quadrant. The area given by [tex] |_a^b x f(x) = b f(b) - a f(a) [/tex] is the area of a big rectangle minus the are of a smaller rectangle. This are includes the area given by [itex] \int_a^b f(x) dx. [/itex]
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 [itex] \int_a^b x f'(x) dx [/itex] 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 [itex] \int_a^b x f'(x) dx [/itex] ? 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.
Last edited by a moderator: