x, y, r, phi = var('x y r phi')
f(x, y) = sign(x^2 + y^2 - 4)
T(r, phi) = [r*cos(phi), r*sin(phi)]
J = diff(T).det().simplify_full()
T_f = f.substitute(x=T[0], y=T[1])
int_f = integral(integral(T_f*abs(J), r, 0, 3), phi, 0, 2*pi).simplify_full
()
show(r"$\iint\limits_\Omega%s = %s$"%(latex(f(x)), latex(int_f)))
Returns correct answer: $\pi$, while
x, y, r, phi = var('x y r phi')
f(x, y) = sign(x^2 + y^2 - 4)
T(r, phi) = [r*cos(phi), r*sin(phi)]
J = diff(T).det() #.simplify_full()
T_f = f.substitute(x=T[0], y=T[1])
int_f = integral(integral(T_f*abs(J), r, 0, 3), phi, 0, 2*pi).simplify_full
()
show(r"$\iint\limits_\Omega%s = %s$"%(latex(f(x)), latex(int_f)))
Yields $-9\pi$
The only thing simplify_full() changes here is it applies identity $\sin^2
+ \cos^2 = 1$
Code was executed in SMC worksheet
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