Check the value of r:
r = 0:0.07
It is a scalar wth value 0 because the colon operator ":" uses an increment of 1 by default. Since 0.07 - 0 < 1, you get just 0.
Perhaps you meant to use a smaller increment:
r = 0:0.01:0.07
Then the rest of the code runs:
w = 0:4;
[R,W] = meshgrid(r,w);
V1 = @(R,W) -acosh(10*(W./(1600*R + 21))^(1/2))/20000000000;
V2 = @(R,W) acosh(10*(W./(1600*R + 21))^(1/2))/20000000000;
fun = @(x,R,W)0.0018./((W./((cosh(10^10.*x./0.5)).^2)-(R.*16+0.21)).^0.5);
q = @(R,W) integral(@(x)fun(x,R,W),V1(R,W),V2(R,W));
Q = arrayfun(q,R,W);
surf(R,W,Q)
