Definite integral of piecewise function | AP Calculus AB | Khan Academy

Sal evaluates definite integral of a piecewise function over an interval that goes through the two cases of the function.

AP Calculus AB on Khan Academy: Bill Scott uses Khan Academy to teach AP Calculus at Phillips Academy in Andover, Massachusetts, and heÕs part of the teaching team that helped develop Khan AcademyÕs AP lessons. Phillips Academy was one of the first schools to teach AP nearly 60 years ago.

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[laureanooliva]

would it be wrong if for the first part of the function I used limit as a approaches 0 to define the upper bound. Because the function is actually never 0, at least not that part of the function. My math teacher told me that was wrong but he couldn't explain why.

[raivatshah]

Hey Sal, I have a question: Wouldn't this only work if the piecewise function is continuous? What about graphs with jump or other discontinuities?

[krishnac]

Hey Sal, for Sin(π) i am getting a value of 0.054 and for Sin(180deg) i'm getting a value of 0 (the computation is done through a scientific calculator). Why is there such a discrepancy in scientific calculator calculation? Can you guide us on this?

[JohnDoe-nrzi]

what would happen if the first function included zero, but not the second one?
like:

f(x) = { x+1 for x<=0, cos(pi*x) for x > 0) }
Would this change anything?