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jedai17
what is double integral function? how to solve double integral function?
Double Integral: http://mathworld.wolfram.com/FubiniTheorem.html You solve in the order of the integral, if dy comes first, you solve as simple integral considering x as constant, after that, solve the integral for dx.
is the value of dy similar to the value of dx?
yes they are similar. when you solve the integral according to dy all x values are constant for you. It doesnt matter which one you solve first, dx or dy.
ahh..ok tnx..can you teach me how to solve this?i have no background in solving this,, \[\int\limits_{0}^{2}\int\limits_{0}^{x ^{2}/2} x/\sqrt{1 + x^2 + y^2} dydx\]
if the limits for both the variables were constant, then we can integrate any variable first., but here since the limits of inner integral is not constant, we need to evaluate the inner integral first, limits are y=0 to y=x^2/2 so, since you are integrating w.r.t 'y', treat x as constant, so you have, \(\int\limits_{0}^{2}x[\int\limits_{0}^{x ^{2}/2} 1/\sqrt{1 + x^2 + y^2} dy]dx\) now the inner integral is od the form, \(\int \dfrac{1}{\sqrt{a^2+y^2}}dy\) do you know how to integrate this ?
i just took 1+x^2 = constant = a^2 because we have a standard integration formula for, \(\int \dfrac{1}{\sqrt{a^2+y^2}}dy\)
then we need to re-substitute back after integration, a^2= 1+x^2 before integrating w.r.t x
*after integration w.r.t y,
in this we are integrating y and we assume as x as constant?
what do you mean by w.r.t y, ?
yes, true, so we'll get answer in 'x' after first integrating w.r.t y (with respect to y)
ok, \(\int \dfrac{1}{\sqrt{a^2+y^2}}dy= (1/a)\arctan(y/a)+c\) but then you have to use integration by parts to integrate w.r.t x. so, i suggest we better CHANGE the ORDER of integration, that is, we'll change the limits so that we can first integrate w.r.t 'x' and then 'y'. For that, wee need to plot 2 regions (for 2 limits): (1) y= 0 to y=x^2/2 (2) x=0 to x= 2 can you plot these 4 curves (3 of these are simple straight lines)?
i'm sorry don't know how to plot this..
i'm sorry i forgot our previous lessons about that..
so, you say you don't know how to plot lines x=0,y=0,x=2 ? i'll help you with y=x^2/2 but if you don't know how to plot those lines, then it'll be very difficult for you to understand this..
i know this x=0,y=0,x=2 but i don't know y=x^2/2..i'm sorry
so, plot these x=0,y=0,x=2 using 'Draw' tool here.
yes, x=0 is y-axis, y=0 is x axis , and x=2 you have drawn correct, now y= x^2/2 is a parabola which has a plot, |dw:1362400675976:dw| so you are integrating in the region that i am gonna shade
why it becomes parabola?
|dw:1362400803859:dw| since we were integrating w.r.t 'y' first, i've drawn vertical lines, now we are changing the order of integration, we need to draw horizontal lines,
|dw:1362400889186:dw| to |dw:1362400935932:dw|
i'm sorry i forgot how to find the equation of the line
oh, its easy, that a horizontal line , so u just need co-ordinates of intersection, which you can find by solving simultaneously, x=2 and x= \(\sqrt{2y}\)
by the way, did you get that final diagram ?
which part you have doubt with ?
\[x =2 and x = \sqrt{2y}\] i don't know how to fnd the equation of this line..
x= 2, x=\(\sqrt{2y}\) so, 2= \(\sqrt{2y}\) can u find 'y' from here ?
:O that was simple arithmetic....sorry to say but, if you have problem solving that you should think before dealing double integration, sorry if i am rude, but you are lacking basics ... 2= \(\sqrt{2y}\) 4= 2y (by squaring) y=2 this is the equation of your horizontal line
yes that's true i almost forgot the topics about integration that's why i am like this..
|dw:1362402005207:dw| so your new limits are, \(\huge \int \limits_0^2\int \limits_{\sqrt{2y}}^2\dfrac{x}{\sqrt{1+x^2+y^2}}dx.dy \) now you have to first integrate w.r.t x to do that put u = 1+x^2+y^2 du=... ?
oh, since we are integrating w.r.t 'x' first now, we treat 'y' as constant, so du =2xdx or x dx = du/2 got this ?
so, your inner integral will become, \(\int \dfrac{du}{2\sqrt u}\) can you integarte this ?
as long as weve gotten this u substitution correct: \[\int \frac{1}{2\sqrt{u}}du=\sqrt{u}\]
assuming that u = 1+x^2+y^2, plug back in for u \[\sqrt{ u}=\sqrt{ 1+x^2+y^2}\] from x=sqrt(2y) to 2 \[\sqrt{ 1+2^2+y^2}-\sqrt{ 1+(\sqrt{2y})^2+y^2}\] \[\sqrt{ 5+y^2}-\sqrt{ 1+2y+y^2}\]
y^2+2y+1 = (y+1)^2 sooo \[\int\sqrt{5+y^2}-(y+1)~dy\]
why it becomes subtraction?
why it become y^2+2y+1 = (y+1)^2 ?
because: \(\Large \int_{a}^{b} f(x)~dx=F(b)-F(a)\)
in this time we are integrating y?
when we replace the \(\sqrt{u}\) with what we defined it to begin with: \(u=1+x^2+y^2\) and input the ab limits of a=\(\sqrt{2y}\) and b=2 we get \[(\sqrt{1+b^2+y^2})-(\sqrt{1+a^2+y^2})\] \[(\sqrt{1+b^2+y^2})-(\sqrt{1+a^2+y^2})\] and yes, then since we are finished with the inside integral, we focus on the next integral
the next integration is next ... \[\int\sqrt{5+y^2}-(y+1)~dy\]
is it correct?i'm sorry have a problem in integration..
the square root part seems a little off to me
and it looks like you tried a derivative onthe y+1 parts instead of integeration
\[-\int y+1~dy=-(\frac{y^2}{2}+y)\]
yes.. i'm sorry a have a problem in integration..
why is that u not integrate \[\sqrt{5 + y^2}\]?
i did not integrate it because I want you to work it out. you are the one that needs the practice at it :)
if you cant see the integration right off hand, try the "table" method ... which is simply looking up the integral in a table of integrals: look for \(\sqrt{a^2+x^2}\)
ah.. wat is this \[\sqrt{a^2 + x^2}\]? how will i use this?
i looked it up in a table, just to be sure. Look at #30
\[\int \sqrt{5+y^2}dy=\frac12y\sqrt{5+y^2}+\frac12 5^2ln(y+\sqrt{5+y^2})\]from y=0 to y=2
the 5^2 ln part is a typo; just 5 ln ...
\[y/2 \sqrt{5 + y^2} + 5/2 \ln (y + \sqrt{5 + y^2}\]
plug in the values of your limit to determine the values
\[(3+\frac52 ln(5))-(\frac54ln(5))-4\] the -4 is from the simpler part we did earlier
with any luck this simplifies to:\[\frac54ln(5)-1\]
is this the answer? why -4 ?
\[\int\sqrt{5+y^2}-(y+1)~dy\] \[\int\sqrt{5+y^2}dy-\int y+1~dy\] \[\int\sqrt{5+y^2}dy-(\frac12y^2+y)\] and since y=0 to 2 \[\int\sqrt{5+y^2}dy-4\]
i cant verify the steps taken to swap the limits from dy dx to dx dy at the moment, but assuming that was done correctly .... we have come to the end of it
i'm sorry..i don't understand it. it's a little bit difficult... :(
the hardest part was not the y^2/2+y ... that just equals -4 we worked thru the sqrt(5+y^2) from the table, and inputted the limit values to get: \[\frac54ln(5)-4\]as a final result, assuming the swapping of dydx to dxdy was correct to begin with
i must study first and review about integral in order for me to understand this..
please teach me again next time..thanks for your time.. :)