Prove \int _a ^b f(x) \: dx with Continuous f

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In summary, the problem states that if f is a one-to-one function with a continuous derivative, then the integral of f(x) from a to b is equal to bf(b) - af(a) - the integral of f^-1(y) from f(a) to f(b). This can be visualized by drawing f(x) over the interval [a,b] and constructing rectangles with areas of a*f(a) and b*f(b). This relation can also be seen in a plot of the function y(x)=0.2x^2 from 2 to 4.
  • #1
DivGradCurl
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Problem:

(a) If [tex]f[/tex] is one-to-one and [tex]f^{\prime}[/tex] is continuous, prove that

[tex]\int _a ^b f(x) \: dx = bf(b) - af(a) - \int _{f(a)} ^{f(b)} f ^{-1} (y) \: dy[/tex]

(b) In the case where [tex]f[/tex] is a positive function and [tex]b > a > 0[/tex], draw a diagram to give a geometric interpretation of part (a).

My work:

(a) [tex]\int _a ^b f(x) \: dx[/tex]

Integrating by parts gives

[tex]u = f(x) \Rightarrow \frac{du}{dx} = f ^{\prime} (x) \Rightarrow du = f ^{\prime} (x) \: dx[/tex]
[tex]dv = dx \Rightarrow v = x[/tex]

[tex]\int _a ^b f(x) \: dx = \left. xf(x) \right] _a ^b - \int _a ^b x f ^{\prime} (x) \: dx[/tex]
[tex]\int _a ^b f(x) \: dx = bf(b) - af(a) - \int _a ^b x f ^{\prime} (x) \: dx[/tex]

Applying the Substitution Rule gives

[tex] y = f(x) \Leftrightarrow x = f^{-1} (y) \Rightarrow \frac{dy}{dx} = f ^{\prime} (x) \Rightarrow dx = \frac{dy}{f ^{\prime} (x)}[/tex]

[tex]y(b) = f(b)[/tex]
[tex]y(a) = f(a)[/tex]

[tex]\int _a ^b f(x) \: dx = bf(b) - af(a) - \int _{f(a)} ^{f(b)} f ^{-1} (y) \: dy[/tex]

(b) I'm not sure how I should handle this one. The left-hand side is quite easy to visualize: it corresponds to a generic integral from a to b. The right-hand side does not seem to be that simple, and I need some help.

Any help is highly appreciated.
 
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  • #2
1) Draw orthogonal x-and y-axes on a piece of paper
2) Mark the interval [a,b] on the x-axis, and draw f(x) over it, so that it ranges between f(a) and f(b); mark f(a) and f(b) on the y-axis.
3) What rectangle can you naturally construct whose area is a*f(a)?
4) What rectangle can you naturally construct whose area is b*f(b)?
5) Look and behold, and see if you find an easy geometric interpretation of the equality..
 
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  • #3
You know Thiago, I don't claim to know much math, no more than 1% in fact, so I'm not surprised your relation is news for me and I'm sure you will follow Arildno so I don't think I'm giving anything away by posting the attached plot for the function [itex]y(x)=0.2 x^2 [/itex] from 2 to 4.
 

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  • #4
I met this relation in my first analysis course, and it was, to me, one of those "Wow, math is really cool!"-experiences..
 
  • #5
Thank you guys! I can now see what you're talking about. It's pretty straight-forward. I was a bit confused by a "big fat" generic equation. :)
 
  • #6
saltydog said:
You know Thiago, I don't claim to know much math, no more than 1% in fact, so I'm not surprised your relation is news for me and I'm sure you will follow Arildno so I don't think I'm giving anything away by posting the attached plot for the function [itex]y(x)=0.2 x^2 [/itex] from 2 to 4.

Just want to correct my statement above: The function [itex]y(x)=0.2x^2[/itex] is NOT one-to-one and only because it's so in the first quadrant would it qualify for the relation above with a and b in the same quadrant.
 

Related to Prove \int _a ^b f(x) \: dx with Continuous f

1. What does it mean to "prove" an integral?

Proving an integral means showing that the value of the definite integral of a function is valid and accurate. This involves using mathematical techniques and principles to verify the solution.

2. Why is it important for the function to be continuous?

A continuous function is one that has no breaks or gaps in its graph. This is important for proving an integral because it ensures that the function is well-behaved and can be integrated using traditional methods. If a function is not continuous, it may require more advanced techniques to prove its integral.

3. What are the key steps to proving an integral?

The key steps to proving an integral include: 1) determining the limits of integration, 2) finding the antiderivative of the function, 3) evaluating the antiderivative at the limits of integration, and 4) subtracting the value at the lower limit from the value at the upper limit to find the final result.

4. Can you use different techniques to prove an integral?

Yes, there are multiple techniques for proving an integral, including the Fundamental Theorem of Calculus, integration by substitution, and integration by parts. The choice of technique will depend on the complexity of the function and the problem at hand.

5. How can I check if my integral proof is correct?

You can check your integral proof by plugging the values of the limits of integration into the antiderivative and verifying that it matches the original function. You can also use a graphing calculator or software to graph both the original function and its integral and make sure they align correctly.

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