Geometric interpretation of \int x f'(x)

In summary, The conversation discusses the geometric visualization of the formula \int_a^b x f'(x), which arises from integration by parts. The area given by |_a^b x f(x) = b f(b) - a f(a) represents a big rectangle minus a smaller rectangle, including the area given by \int_a^b f(x) dx. The term \int_a^b x f'(x) dx subtracts the excess area formed by horizontal line segments under the curve of f(x) between the lines y=f(a) and y=f(b). This interpretation may be useful for teaching, but it is not commonly found in textbooks. The importance of understanding integration by parts as the Leibniz rule of
  • #1
Stephen Tashi
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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.
 
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  • #2
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.
 

What is the geometric interpretation of ∫ x f'(x)?

The geometric interpretation of ∫ x f'(x) is the area under the curve of the function f'(x) between the limits of integration, x = a and x = b, where a and b are the starting and ending points on the x-axis. This area represents the net change in the function f(x) over the interval [a, b].

How is the geometric interpretation of ∫ x f'(x) related to the Fundamental Theorem of Calculus?

The Fundamental Theorem of Calculus states that the derivative of the integral of a function is equal to the original function. Therefore, the geometric interpretation of ∫ x f'(x) is related to the Fundamental Theorem of Calculus because it represents the net change in the function f(x) over an interval, and the derivative of this integral is equal to the original function f'(x).

What is the significance of the sign of ∫ x f'(x)?

The sign of ∫ x f'(x) indicates whether the net change in the function f(x) over the interval [a, b] is positive or negative. If the integral is positive, it means that the function is increasing over the interval, and if it is negative, it means that the function is decreasing over the interval.

Can the geometric interpretation of ∫ x f'(x) be extended to higher dimensions?

Yes, the geometric interpretation of ∫ x f'(x) can be extended to higher dimensions. In two dimensions, it represents the area under a curve on a plane, and in three dimensions, it represents the volume under a surface in space. This concept can be further extended to multiple dimensions, known as multivariate calculus.

How can the geometric interpretation of ∫ x f'(x) be applied in real-world situations?

The geometric interpretation of ∫ x f'(x) can be applied in real-world situations to calculate the net change in a quantity over a given interval. For example, it can be used to calculate the total distance traveled by an object with varying velocity over a certain time period, or the total sales of a company over a specific time period with changing revenue rates.

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