Calculating integration limits of a double integral

Hello,

I am working on a college assignment to calculate the symbolical value and the numerical approximation of the double integral of over the region defined by and to determine the integration limits in y by symbolical computation.

Unfortunately, I haven't been able to figure out how to calculate the integration limits without knowing the final result. Sorry if this is easy, but I am a beginner in Matlab and my knowledge of double integrals isn't very profound.

Could someone please give me a hint how to solve this problem?

Thank You.

[Alan Stevens]

Do you mean

x^2/6 + y^2/4 = 1     % you wrote x^2/6 + x^2/4 = 1

If so, this is the equation of an ellipse cemtred on (0,0) with semimajor and semiminor axes of length sqrt(6) and 2. You can rearrange the equation to get y = in terms of x, then you have four quadrants to consider for the appropriate sign of sin(x*y).

[Chuguang Pan]

The result is 0 ?

[Alan Stevens]

It looks like it! But don't rely on my say so, do the calculations.

[mj702]

Yes, exactly! The second x should have been an y. Sorry for the mistake.

Thank You

[Bruno Luong]

x^2/6 + y^2/4 = 1 is a curve (ellipse). It's 1D manifold. I don't know hat "double integral" is doing here. You probably want a curviline integral.

[Bruno Luong]

The integral is 0 since the function is odd in x

f(x,y)=-f(-x,y)

and the domain is symetric in x.

(x,y) in domain then (-x,y) is in.

You can deduce the same conclusion with y variable.

Answer 1

[Rohit Pappu]

For computing the numerical approximation of the above double integral

%% Define the major and minor axis of ellipse
a = sqrt(6);
b = sqrt(4);

%% Define f(x,y)
fun = @(x,y) sin(x.*y);

%% Convert f(x,y) into f(r,theta) 
polarfun = @(theta,r) fun(a*r.*cos(theta), b*r.*sin(theta))*a*b.*r;

%% x = a*r.*cos(theta), y = b*r.*cos(theta) , dxdy = a*b*r*dr(dtheta)

%% Assign upper limit for r
rmax =1;

%% Calculate the integral for r = [0,1] and theta = [0,2*pi]
q = integral2(polarfun,0,2*pi,0,rmax)

Additional documentation about integral2 can be found here