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jennychan12Best ResponseYou've already chosen the best response.0
the piecewise function f(x) = \[\frac{ x^2 }{ \left x \right }, x \ne 0\] \[0, x=0\] find \[\int\limits_{1}^{4} f(x)dx\] how would u do that?
 one year ago

scarydoorBest ResponseYou've already chosen the best response.1
Riemann integrals are not affected by singular points. So you can do the integral as though the point x=0 doesn't exist.
 one year ago

scarydoorBest ResponseYou've already chosen the best response.1
Plus, the point x=0 doesn't even come up in the integral, since you're taking the integral from 1 to 4.
 one year ago

scarydoorBest ResponseYou've already chosen the best response.1
and so then you know also that x > 0, so on the integral (1,4) f(x) = x^2 / x. (explicitly: you can take away the absolute value sign)
 one year ago

jennychan12Best ResponseYou've already chosen the best response.0
so basically \[\int\limits_{1}^{4} \frac{ x^2 }{ \left x \right } dx\] ?
 one year ago

scarydoorBest ResponseYou've already chosen the best response.1
Yeah that's right. This is intuitively why singular points don't really matter: In this picture, there's a singular point where the function is different at just one point. But, you remember when you do the Riemann approximation of the integral (the area under the curve) how you multiply the value of the function by the width? Well Singular points basically have no width... so the area underneath them, is effectively nothing. So you can ignore them. dw:1355969105016:dw
 one year ago

jennychan12Best ResponseYou've already chosen the best response.0
but what about the absolute value?
 one year ago

scarydoorBest ResponseYou've already chosen the best response.1
What you can do for the absolute value, at least in this case, is figure out whether x, over the interval of integration, is positive or negative, and then use that knowledge to remove the absolute value sign.
 one year ago

scarydoorBest ResponseYou've already chosen the best response.1
For example, you are integrating from x=1 to x=4. You know that x is positive for all of those values. So you know that across that whole integral \[ x =  x  \]
 one year ago

scarydoorBest ResponseYou've already chosen the best response.1
And so where you see \[ x  \] you can just write \[ x \]
 one year ago

jennychan12Best ResponseYou've already chosen the best response.0
wait. the abs. value doesnt matter cuz it's always positive from 14 right?
 one year ago

jennychan12Best ResponseYou've already chosen the best response.0
\[\int\limits_{4}^{2} f(x)dx\]
 one year ago

jennychan12Best ResponseYou've already chosen the best response.0
wait. but i think it's the same concept
 one year ago

scarydoorBest ResponseYou've already chosen the best response.1
Ah. okay. Well, you have something like this: dw:1355970713671:dw Where the integral is summing up that "shaded" area. So you can write it like this: \[\int\limits_{4}^{2}f(x) dx = \int\limits_{4}^{0} f(x) dx + \int\limits_{0}^{2} f(x) dx\]
 one year ago

jennychan12Best ResponseYou've already chosen the best response.0
but how would you integrate that? just assume it's x^2/x ??
 one year ago

scarydoorBest ResponseYou've already chosen the best response.1
That, in effect, is just breaking up the integral into two more. Now you know that from x=4 to x=0 that x <= 0, so that \[x = x \]. And on x=0 to x=2 \[ x = x \]
 one year ago

jennychan12Best ResponseYou've already chosen the best response.0
oh wait nvm i got it
 one year ago

scarydoorBest ResponseYou've already chosen the best response.1
I got 10. I might have been a little rough in calculation though.
 one year ago

scarydoorBest ResponseYou've already chosen the best response.1
wolfram alpha confirms an answer of 10.
 one year ago

jennychan12Best ResponseYou've already chosen the best response.0
stupid negative sign... either way, the answer's 2
 one year ago

jennychan12Best ResponseYou've already chosen the best response.0
ohh can i see the wolfram alpha one? do u have a link?
 one year ago

scarydoorBest ResponseYou've already chosen the best response.1
http://www.wolframalpha.com/input/?i=integral+from+x+%3D+4+to+x%3D2+of+x^2+%2F+abs%28x%29+dx
 one year ago

scarydoorBest ResponseYou've already chosen the best response.1
\[\int\limits_{4}^{2} \frac{x^2}{x} dx = \int\limits_{4}^{0} \frac{x^2}{x} dx + \int\limits_{0}^{2} \frac{x^2}{x} dx\]
 one year ago

jennychan12Best ResponseYou've already chosen the best response.0
oh just that ok then yeah i understand
 one year ago

scarydoorBest ResponseYou've already chosen the best response.1
\[= \int\limits_{4}^{0}  x dx + \int\limits_{0}^{2} x dx\]
 one year ago

jennychan12Best ResponseYou've already chosen the best response.0
but what about when x = 0?
 one year ago

scarydoorBest ResponseYou've already chosen the best response.1
Oh yeah just use limits like the other person did. And it'll come out to what I wrote up there.
 one year ago

scarydoorBest ResponseYou've already chosen the best response.1
\[\lim_{b \rightarrow 0^{}}\int\limits_{4}^{b} \frac{x^2} {x}dx = \lim_{b \rightarrow 0^{}}\int\limits_{4}^{b} \frac{x^2}{x} dx\]
 one year ago

scarydoorBest ResponseYou've already chosen the best response.1
You'd probably write it like that if you wanted to be heaps formal.
 one year ago

jennychan12Best ResponseYou've already chosen the best response.0
just kidding my teacher printed all the answers wrong _
 one year ago

scarydoorBest ResponseYou've already chosen the best response.1
huh? Kidding about what?
 one year ago
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