A problem with evaluating an integral

[mahmoud2011]

$\int^{\pi}_{0} f(x) dx$ where ,

f(x) = sin x if $0 \leq x < \frac{\pi}{2}$
and f(x) = cos(x) if $\frac{\pi}{2} \leq x \leq \pi$

The problem is that the version I am using of Fundamental theorem is if f is continuous on some closed interval , I wrote the integral as

$\int^{\pi / 2}_{0} f(x) dx + \int^{\pi}_{\pi /2} f(x) dx$

but I have in the first integral f still is not continuous on $[0,\pi/2]$

I tried to open some references and reached another version for the theorem there f is integrable on f , and g' =f , but I couldn't do any thing

Thanks

 

[Erland]

but I have in the first integral f still is not continuous on $[0,\pi/2]$

That's no problem. This integral is the same as the integral of sin x over all of [0,pi/2], since the two functions are equal except in a single point. And if you change an integrable function in a single point (or finitely many points) then the function is still integrable with the same integral. This can be seen by using upper and lower sums.

 

[mahmoud2011]

The problem is that the textbook I am using doesn't introduce the integrals by first introducing the upper Darboux sums and the lower Darboux sum , it uses another approach which introduce the area as the limit of sums of approximating rectangles.

 

[mahmoud2011]

Thanks , Now I am reading in another textbook Which deals with this point precisely .

 

[mahmoud2011]

That's no problem. This integral is the same as the integral of sin x over all of [0,pi/2], since the two functions are equal except in a single point. And if you change an integrable function in a single point (or finitely many points) then the function is still integrable with the same integral. This can be seen by using upper and lower sums.

now I justified my solution as following , the first definite integral exist because f is integrable on [0,pi] (since it is piece-wise continuous ) , now since f is a function on [0,pi\2] where f= sin (x) ,0 ≤x≤ pi/2 , except at pi/2 , their definite integral is the same and then we can use the FTC , or I can use the version of the theorem which apply to any integrable function which is not necessary continuous we know that f is integrable on [0,pi/2] so we want to find a continuous function g such that g' = f on (0,pi/2) and g is continuous on [0,pi/2] so we see that this is g is sin (x) , ( I think all of this will be easy in practice ) , but what I am wandering of is why this is an exercise in my text-book , although it doesn't concentrate at this point yet .

Thanks

 

[Erland]

Yes, use FTC to sin x over [0,pi/2] and observe that the given function has the same integral over this interval since it differs only at one point.

I think your textbook should have mentioned thia problem,

 

[mahmoud2011]

Yes, use FTC to sin x over [0,pi/2] and observe that the given function has the same integral over this interval since it differs only at one point.

I think your textbook should have mentioned thia problem,

no , it hasn't mentioned it yet , it uses a definition of integral which depends on dividing the interval into sub intervals equal in length , so I began to read a bout the theory of integral in Advanced Calculus by agnus Taylor is this good for me .

 

[mahmoud2011]

So after I have read Integration it some Advanced Calculus books , I had only one question to ask , this only to confidence myself with only a part of of a proof ( you Know that the proof in Advanced books depend leaving details for you ) , it wrote that if f is integrable , then there exist partitions P' and P'' , such that given ε >0
U(f,P') < U(f) + ε/2 and L(f,P'') > L(f) -ε/2

Here I began to justify this to my self as following , I would only now justify the first the second one is similar , we will assume it is not true, then we have

U(f,P') ≥ U(f) + ε/2 > U(f)

and this contradicts that U(f) is the grates lower bound of of all Darboux sums
But I see that this justification is not good at all because of there is P' such that U(f,P') = U(f) + ε/2 , so I began to make another one which I see is a better one assume that
U(f,P') > U(f) + ε/4 , then this gives a contradiction , so We have

U(f,P') ≤ U(f) + ε/4
and since ε/4 < ε/2
so we have U(f,P') ≤ U(f) + ε/4 < U(f) + ε/2

Is all of What I have said is true or not



Thanks