Olivia B.

asked • 12/01/16

How to find values of a function based off of it's definite integral (am I thinking about this wrong, or do I just plug in the number?)

F(x)= ∫0 to x sin(2t) dt
 
evaluate F(pi)
For this one I thought I would just plug in sin(2pi) but I got that wrong. It asks for an exact number so I thought I shouldn't give the decimal answer. Is there something I overlooked?
 
and for what interval is F(x) non- negative?
for this one I found the derivative and based on that said from [0, infinity) it would be positive but I got it wrong.

2 Answers By Expert Tutors

By:

Arturo O. answered • 12/01/16

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Olivia B.

Okay so I have to take the derivative first and then plug in pi? Why did you do that? is that what the dt notation signifies to do before the value is plugged in?
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12/01/16

Arturo O.

You integrate first and then plug in π.  You just perform the definite integration the usual way, but since x is a variable, your definite integral will be an expression containing x, not a number (i.e. not a final number like when you integrate between two given limits).  Then you plug in a specific value for x.  
 
If you plug in π before integrating, you get zero.
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12/01/16

Olivia B.

Oh. Okay, thank you! This means I should then find the interval after integration as well which would make the non positive values (-infinity to infinity)? Thank you for your help!
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12/01/16

Arturo O.

From
 
F(x) = [1 - cos(2x)]/2
 
you can see that
 
F(x) ≥ 0
 
as long as cos(2x) ≤ 1.
 
But cos(2x) is always ≤ 1, so it looks like x can be anywhere in (-∞,∞), as you suggested.
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12/01/16

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