Variable in function as well as integral boundary

Hi,

I have an integral I want to compute and plot for different angles theta. 0-90 degrees.

For every angle I want to compute the integral of the function from 0 up to the angle theta.

theta = 1:90;
A = 0.4*10^-19;
R = 4215 * theta.^(-1.35)*10^-9;
D_0 = 0.3;
D_c = R.*(1-cosd(theta));
D = (D_0 + D_c)*10^-9;
b = 0.2*10^-6;

Fun = @(theta) (A ./ (6 .* pi .* (D_0 + R .* ( 1-cosd(theta))).^3)) .* b .* R;

F_vdW1 = integral(Fun, 0, theta);

Error using integral
Limits of integration must be double or single scalars.

Can someone please help with what I am doing wrong?

Thanks

Answer 1

[Torsten]

Is it this what you want ?

If not, please clarify how the theta in the integrand and in the upper bound of the integral should be interpreted. Mathematically, your notation to use the same name for both is wrong.

theta = 1:90;
A = 0.4*10^-19;
R = 4215 * theta.^(-1.35)*10^-9;
D_0 = 0.3;
D_c = R.*(1-cosd(theta));
D = (D_0 + D_c)*10^-9;
b = 0.2*10^-6;
Fun = (A ./ (6 .* pi .* (D_0 + R .* ( 1-cosd(theta))).^3)) .* b .* R;
F_vdW1 = cumtrapz(theta,Fun);

F_vdW1 = 1×90

1.0e-30 * 0 0.0461 0.0666 0.0793 0.0881 0.0948 0.1002 0.1046 0.1083 0.1115 0.1143 0.1167 0.1189 0.1209 0.1227 0.1243 0.1258 0.1272 0.1285 0.1297 0.1308 0.1319 0.1329 0.1338 0.1347 0.1355 0.1363 0.1371 0.1378 0.1385

plot(theta,F_vdW1)

[Paul Heineman]

I want to do this:

Where phi is theta

[Torsten]

Are you sure that "phi" should enter the equation for R in degrees and not in radians ?

And according to your graphics, only the 6 and not the complete expression 6*pi*[D0+R*(1-cos(phi))]^3 would appear in the denominator of the integrand.

A = 0.4*10^-19;
R = @(phi) 4215 * phi.^(-1.35)*10^-9;
D_0 = 0.3;
D_c = @(phi) R(phi).*(1-cosd(phi));
D = @(phi) (D_0 + D_c(phi))*10^-9;
b = 0.2*10^-6;
Fun = @(phi)(A ./ (6 .* pi .* (D_0 + R(phi) .* ( 1-cosd(phi))).^3)) .* b .* R(phi);
theta = 1:360;
F_vdW1 = arrayfun(@(theta)integral(Fun,0,theta),theta)

F_vdW1 = 1×360

1.0e-28 * 0.6801 0.5336 0.4630 0.4187 0.3872 0.3633 0.3442 0.3285 0.3152 0.3038 0.2938 0.2850 0.2771 0.2700 0.2636 0.2577 0.2523 0.2473 0.2427 0.2384 0.2343 0.2305 0.2270 0.2236 0.2204 0.2174 0.2146 0.2119 0.2093 0.2068

plot(theta,F_vdW1)

[Paul Heineman]

This is what I needed, thanks!

Answer 2

[Bruno Luong]

I have no idea if the code correspondons to the formula; I just modify your code to make it work on array

theta = 1:90;
A = 0.4*10^-19;
R = @(theta) 4215 * theta.^(-1.35)*10^-9;
D_0 = 0.3;
D_c = @(theta) R.*(1-cosd(theta));
b = 0.2*10^-6;

Fun = @(theta) (A ./ (6 .* pi .* (D_0 + R(theta) .* ( 1-cosd(theta))).^3)) .* b .* R(theta);
F_vdW1 = arrayfun(@(onetheta) integral(Fun, 0, onetheta), theta)

F_vdW1 = 1×90

1.0e-28 * 0.6801 0.5336 0.4630 0.4187 0.3872 0.3633 0.3442 0.3285 0.3152 0.3038 0.2938 0.2850 0.2771 0.2700 0.2636 0.2577 0.2523 0.2473 0.2427 0.2384 0.2343 0.2305 0.2270 0.2236 0.2204 0.2174 0.2146 0.2119 0.2093 0.2068