Question

what is double integral function?
how to solve double integral function?

Answer 1

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.

Answer 2

is the value of dy similar to the value of dx?

Answer 3

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.

Answer 4

ahh..ok tnx..can you teach me how to solve this?i have no background in solving this,,

\[\int\limits_{0}^{2}\int\limits_{0}^{x ^{2}/2} x/\sqrt{1 + x^2 + y^2} dydx\]

Answer 5

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 ?

Answer 6

why it become a^2?

Answer 7

i just took 1+x^2 = constant = a^2
because we have a standard integration formula for, \(\int \dfrac{1}{\sqrt{a^2+y^2}}dy\)

Answer 8

then we need to re-substitute back after integration, a^2= 1+x^2 before integrating w.r.t x

Answer 9

*after integration w.r.t y,

Answer 10

in this we are integrating y and we assume as x as constant?

Answer 11

what do you mean by w.r.t y, ?

Answer 12

yes, true, so we'll get answer in 'x' after first integrating w.r.t y (with respect to y)

Answer 13

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)?

Answer 14

i'm sorry don't know how to plot this..

Answer 15

i'm sorry i forgot our previous lessons about that..

Answer 16

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..

Answer 17

i know this x=0,y=0,x=2 but i don't know y=x^2/2..i'm sorry

Answer 18

so, plot these x=0,y=0,x=2 using 'Draw' tool here.