Double Integral

Iancar29

Hi guys , just trying to work through some double integration examples and
still cant get my head around them at how to tackle the more complex ones.

Q

0 2√(1-x^2)
∫ dx ∫ x dy
-1 0

Once in order x, y & then in order y,x .

Its the above line that gets me confused , like

Any help with how to start tackling it much appreciated .

sponsoredwalk

Assuming the integral is [latex]\int_{-1}^0 \int_0^{2\sqrt{1-x^2}}x dy dx[/latex] you first draw a graph of the domain you're integrating over, which in this case is a segment of an ellipse.

Then you read it from outside to inside as "for a given value of x, y goes from 0 to [latex]2\sqrt{1-x^2}[/latex]", so keeping any x values in the function constant (in this case [latex]f(x,y) = x[/latex]) we integrate w.r.t. y only, then after evaluating integrate w.r.t. x.

Looking at it this way it's easy to see that there's no reason why we need to integrate w.r.t. x first instead of y, why not just change the order of integration since it ends up covering the same domain? Thus, to reverse the order of integration you just look at your picture and ask, for a given value of y how do the x values change?

https://www.youtube.com/watch?v=rkvwEkM5RVo

That's the general idea, graph the domain & ask "for a given value of *, how does ** change?", it gives you great freedom in doing these integrals, but you can only change the order when Fubini's theorem holds (I'll find an intuitive easy proof of it if you'd like).

If you write it as [latex]\int_{-1}^0 dx \int_0^{2\sqrt{1-x^2}}x dy [/latex] then you read from left to right as "for a given value of x, y changes from blah to blah as we integrate [latex]f(x,y) = x[/latex]".

Iancar29

Sorry for the late reply sponsoredwalk ,

was concentrating on other hand ups the past week .

Thank very much for the help! , Shall give it another go now!