anonymous
  • anonymous
Find the area above x-axis for given function on given interval by using the Fundamental theorem of Calculus. F(x)= 1+2sinx on [0,pi/2]
Mathematics
  • Stacey Warren - Expert brainly.com
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katieb
  • katieb
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anonymous
  • anonymous
@DanJS
anonymous
  • anonymous
which ftc should I use first or the second?
DanJS
  • DanJS
is this just to integrate with respect to x over that interval?

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DanJS
  • DanJS
F(upper)-F(lower)
anonymous
  • anonymous
I don't think so. I believe it has already integrated considering it uses an upper case F(x).. \[F(x)=\int\limits f(x)dx\]Probably just plug in the limits
DanJS
  • DanJS
from 0 to half pi, sin(x) is always a positive number, the function is always positive
DanJS
  • DanJS
do the integral of the function from 0 to half pi
anonymous
  • anonymous
find the area above x-axis- (for given function on given interval)
DanJS
  • DanJS
F(x) is the given function, the area under that is the definite integral value for the interval
anonymous
  • anonymous
I wasn't sure if the notation was specific. I was taught that it was, but I guess it may not be.
DanJS
  • DanJS
yeah that is what integrating will do, it can give you an exact area value for spaces that are curved in any way on the boundries
anonymous
  • anonymous
it has closed brackets towards it so I guess just teach me the way u is right I will check about that with my prof tmr
DanJS
  • DanJS
you see the pictures of 5 boxes, 10 boxes, infinite boxes, to estimate the area under curves
anonymous
  • anonymous
Yeah, I'm aware. But I was afraid that the question already presumes that it was integrated and just wants the student to plug in the limits, hence the "using fundamental theorem of calculus" part.
anonymous
  • anonymous
If notation is specific, then \[\large \int\limits f(x)dx=F(B)-F(A)\] I feel like we already have the left hand, now we just need to plug in the limits. But I could be wrong.
anonymous
  • anonymous
you see the pictures of 5 boxes, 10 boxes, infinite boxes, to estimate the area under curves- yes, I know that
anonymous
  • anonymous
but what are u trying to say I am confused now
DanJS
  • DanJS
the given function is F(x) = 1 + 2sin(x), they want the area between x-axis and that function. That is what the fundamental theorem will do, you need to integrate, Just make the next F even larger. haha
anonymous
  • anonymous
X)
anonymous
  • anonymous
okay by using the first fundmental theorem?
DanJS
  • DanJS
\[\int\limits_{0}^{\pi/2}F(x)dx \]
anonymous
  • anonymous
that's the second i think
DanJS
  • DanJS
oh, you talking about that derivative of an integral , other thing?
anonymous
  • anonymous
no the ftc- I think the π/2F(x)dx u gave above is the second fundamental theo of calc
anonymous
  • anonymous
The second fundamental theorem of calculus is: \[F(x)=\int\limits_{0}^{x}f(t)dt \implies \frac{ d }{ dx }[F(x)]=f(x)\]
anonymous
  • anonymous
@DanJS okay what is next plz explain me in terms of math I get confused when u talk in sentences.
anonymous
  • anonymous
X)
DanJS
  • DanJS
Integrate the function and you get x - 2*cos(x) then you take the upper bound and the lower bound and evaluate those in that x-2cos(x) take the value of the upper - value of the lower, that is the answer
DanJS
  • DanJS
|dw:1447318077380:dw|
anonymous
  • anonymous
okay i get the upper bound and lower boun dstuff but how did u got Integrate the function and you get x - 2*cos(x)?
DanJS
  • DanJS
|dw:1447318108190:dw|
DanJS
  • DanJS
there are properties for the integrals, similar to the way you do limits, i think so you can re-write addition of terms, as the addition of 2 integrals
DanJS
  • DanJS
it would be the integral of 1 + integral of 2 sinx then you can move the constant 2 out front the second if you want
DanJS
  • DanJS
remember those limit properties
anonymous
  • anonymous
okay I will look into my notes for that. Plz keep doing the problem after finding the lower and upper bound.
DanJS
  • DanJS
http://www.sosmath.com/calculus/integ/integ02/integ02.html
DanJS
  • DanJS
|dw:1447318458871:dw|
DanJS
  • DanJS
remember cos(pi/2) = 0 and cos(0) = 1 you get an answer of pi/2 + 2
anonymous
  • anonymous
1.571 +2
DanJS
  • DanJS
i would leave it as exact value
anonymous
  • anonymous
or pi/2 +2 yeah that's what i got
anonymous
  • anonymous
okay so the upper and lower bound- u said that it would be (1+2sinx)dx and the other (x-2cosx) I am confused on the upper/lower bound stuff that u said
DanJS
  • DanJS
oh, just the ends of the interval
DanJS
  • DanJS
you are only considering the function over a restricted bounded region 0 to half pi
anonymous
  • anonymous
oh okayy
DanJS
  • DanJS
the work required is just what is in those draw boxes really, less you need to show more integrating properties, since that is the topic you are learning now
anonymous
  • anonymous
right I understand! Okay thank you. There is another one with the same question but it's askin gfor F(x)= 1-x^2 on [0,2] I will open that into a new question if u wan t
DanJS
  • DanJS
maybe show that the function is continuous and defined for all values on the interval, all the requirements for integrating
DanJS
  • DanJS
nah that's ok
anonymous
  • anonymous
thank you so much!
DanJS
  • DanJS
1-x^2 is a parabola vertex at (0,1) and going down you will have some area above and some below the x axis,
DanJS
  • DanJS
yeah, maybe show the function is always positive in that interval
DanJS
  • DanJS
state at the end, that the value is the area they want, not sure, depends on who will be reading it
DanJS
  • DanJS
i am falling asleep.. hah it is after 4am here
DanJS
  • DanJS
gonna take off for now, prolly be on tomorrow
anonymous
  • anonymous
how do i show that the fuction is always positive? yeah we will do the next one tmr lets just finish the first one
anonymous
  • anonymous
yeah that's okay we can do that one tmr but like u were mentioning- to show it's positive how do I do that?
DanJS
  • DanJS
sin(x) is + for the whole first quadrant of angles, 0 to pi/2
anonymous
  • anonymous
yeah I know but where do i put that or hwo do i put that in the work I hae so far?
DanJS
  • DanJS
the second one is the same process integrate 1-x^2 from 0 to 2, the thing is continuous and defined over that interval
DanJS
  • DanJS
[x - (1/3)x^3] from 0 to 2
anonymous
  • anonymous
okay. we can do that one tmr it's okay
DanJS
  • DanJS
k, cya
anonymous
  • anonymous
thanks :) bye tc
anonymous
  • anonymous
the second one is the same process integrate 1-x^2 from 0 to 2, the thing is continuous and defined over that interval [x - (1/3)x^3] from 0 to 2 okay, how did u got [x - (1/3)x^3]? @DanJS
DanJS
  • DanJS
You can integrate each term , a property of integrals \[\large \int\limits_{0}^{2}1*dx - \int\limits_{0}^{2}x^2dx\]
DanJS
  • DanJS
oh yeah i forgot, i remembered i have something, one sec
anonymous
  • anonymous
okay I understnad that now
anonymous
  • anonymous
but from that how did u got [x - (1/3)x^3
DanJS
  • DanJS
summary of lots of things, some arent covered till next semester though, calc 2
anonymous
  • anonymous
wow let me print that and put in in ma binder :)
DanJS
  • DanJS
integration is reverse of differentation if you have a function f(x) , you can find the derivative f ' , the integral will move you the other direction , from f ' to the function f
DanJS
  • DanJS
the integral of 1 is the thing that when you take its derivative is 1, kind of like reverse of th epower rule for derivatives, 1 is a constant, or you can say x^0 power what do you take the derivative of to get x^0 or 1? x, the integrating rule thing you can remember is one of those handful of common integrals, there are a bunch listed on page 1 of that pdf
DanJS
  • DanJS
integrating a constant is the first one,
anonymous
  • anonymous
it says kx+c if you are taking about the cconstact
DanJS
  • DanJS
right if you integrate some number with respect to a variable, you can slide in that variable to the zero power as multiplied on i guess, x^0 = 1 using the power rule, (second on that list), the integral of 1*x^0 = (1/1) * x^1 or just X
DanJS
  • DanJS
so integrating 1, results in X
DanJS
  • DanJS
or any other constant k, number
DanJS
  • DanJS
just remember to increase power of x by 1, from 0 to 1, the x appears
DanJS
  • DanJS
it will become just like doing derivatives of things like 3x^2, you just remember to multiply by the power and reduce it by 1, the power rule of derivatives.. after a bit, you just do the pattern
anonymous
  • anonymous
okay so 1-x^2 how did u put that in context ans got -----> x - (1/3)x^3???
DanJS
  • DanJS
integrating x^n, you just increase the power by 1 and divide by that same value integral x^4 ----> a fifth of xto the fifth
anonymous
  • anonymous
wait am reading what u sai d to understna d
anonymous
  • anonymous
so x^2 would be x^3
DanJS
  • DanJS
for the 1 in the prob, you can just remember for constants to add an x when integrating integral of x^0 ---->x/1 = x
DanJS
  • DanJS
To integrate a power term, it is the second common integrals in the chart you just increase the power by 1, and also divide by that same value, 1 more than the power
DanJS
  • DanJS
\[\huge \int\limits x^n dx = \frac{ x^{n+1} }{ n+1 }\]
anonymous
  • anonymous
x^3/4 correct?
anonymous
  • anonymous
sorry divided by 3
DanJS
  • DanJS
The first part of the integral in the problem is 1 you can also think of that as the same as x^0
DanJS
  • DanJS
integrating x^0 following that rule, you get x / 1, just an x
anonymous
  • anonymous
right I got it now
DanJS
  • DanJS
dont really need to include the integration constants in definite integrals,
DanJS
  • DanJS
they will fall out at the end anyways
anonymous
  • anonymous
just to put everything togather can u show all the steps mathametically to show how u got x - (1/3)x^3 u got x from the "1" and how is (1/3)x^3? we got x^3/3 how did it turned into (1/3)x^3?
DanJS
  • DanJS
those are both the same
anonymous
  • anonymous
oh okay! Okay I finally got that part
DanJS
  • DanJS
i just put the n-1 under the variable, or you can make it a fraction and multiply 1/(n+1)
DanJS
  • DanJS
you can see why you have to divide, look at x^5 the derivative is 5x^4 the integral of that should take you back to x^5
anonymous
  • anonymous
yeah i see it now. Okay
DanJS
  • DanJS
you have to divide by that 5, and also increase the power
DanJS
  • DanJS
i started remembering the integrals of the trig functions, by just thinking of it the other way, thinking what was differentiated to get this thing we have to integrate
DanJS
  • DanJS
like the integral of sin(x), is whatever you differentiate to get sin(x),
anonymous
  • anonymous
okay
anonymous
  • anonymous
but hmm just explain me everything in math/ the equation u had with "n" that's much clear instead of started remembering the integrals of the trig functions, by just thinking of it the other way, thinking what was differentiated to get this thing we have to integrate
anonymous
  • anonymous
so after that we get x - (1/3)x^3 right! @DanJS
DanJS
  • DanJS
ok, yes that is right, you can include the integrating constants if you want, they will cancel at the end when you evaluate the values
anonymous
  • anonymous
and my answers is x-(1/3)x^3
DanJS
  • DanJS
yes, it is from 0 to 2, to get the areas
anonymous
  • anonymous
so there were basically 3 steps for thsi answers and I will include what we discussed about getting x-(1/3)x^3 asmy answer in my notes
DanJS
  • DanJS
|dw:1447373372714:dw|
DanJS
  • DanJS
|dw:1447373499951:dw|
DanJS
  • DanJS
-2/3
DanJS
  • DanJS
The definite integral gives you the NET area between f(x) and the x-axis... with the area above axis positive, and area below axis negative
anonymous
  • anonymous
okay so after finding x-(1/3)x^3 u plug in 2 and 0 as x and find the final answer -2/3. okay that's what e did in the previous one
DanJS
  • DanJS
yeah the fundamental theorem thing \[\int\limits_{a}^{b}f(x)dx = F(b) - F(a)\]
anonymous
  • anonymous
yes i get it now tysm for taking ur time. but tbh I really do not understand when someoen explains math in sentences to me I only learn when it is explained directly. but tysm for explaining makes better sens enow tysm
DanJS
  • DanJS
i have to take off for a hour or so in a minute
DanJS
  • DanJS
ok, i will do that, i just hate typing in the commands.. i will from now on... just was faster typing words.. lol sorry bout that
anonymous
  • anonymous
yeah we are done with this question so u are good to go for now lol
DanJS
  • DanJS
:) , k , be back later
anonymous
  • anonymous
bye thanks

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