You got 2 things wrong (i think) . the error you currently get is for second one. but you will get the error for first one eventually.
first, upper limit of integral shouldn't be function of variable a. because in that case the result of integral would be function of a.
and the other problem is in integral2 function the integration over y (second argument) is first integration so you should change the orders.
syms r theta;
a = 1; %%%%%%%%%%%%%%%%%%%%%%%%
theta_u = -(pi/2);
theta_v = pi/2;
r_u = 0;
r_v = @(theta) (a*cos(theta));
y = @(theta,r) (4 * r.^2 .* cos(theta));
integral2(y,theta_u,theta_v,r_u,r_v)
