Im trying to do some homework in which they gave me a domain and a double integral.
I was trying to use wolfram alpha or symbolab to know how to do it but I can't get them to work, with wolfram I get to draw the domain but I dont know how to integrate with that domain, because I have the integral, which is ∫∫D( x2 + 5y2 )dxdy and the D is the Domain I have in the plot, so Im trying to search some way to tell wolfram you have this plot and you need to calculate this integral in this domain, but I don't know how.
2 Answers
To get WolframAlpha to integrate the function x^2+5y^2 over the domain you can use WolframAlpha notation to integrate over the outer semicircle and subtract the integral over the inner semicircle
integrate x^2+5 y^2, {x,-4,4},{y,0,sqrt(16-x^2)} - integrate x^2+5 y^2, {x,-2,2},{y,0,sqrt(4-x^2)}
If calculations like this exceed either the line length limit or the computation time limit then you might do these as two separate calculations and then subtract the results as a final step.
In:
a = -10;
b = 10;
f = x^2 + 5 y^2;
Integrate[f, x, y] (*Indefinite Integral*)
Integrate[f, {x, a, b}, {y, a, b}] (*Definite Integral*)
Plot3D[f, {x, a, b}, {y, a, b}](*Plot*)
Out:
(x^3 y)/3 + (5 x y^3)/3
80000
