# Calculate the required area

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Nikodin Sedlarevic
on 30 Apr 2022

Edited: Riccardo Scorretti
on 30 Apr 2022

I would like to paint a building (all 5 surfaces) that has 4 vertical straight walls, the upper surface (roof) is described by the function z = g (x, y) for (x, y) ∈ [0.0,4.40] × [0.0, 4.20]. The lower edge of the building is at a height of z (x, y) ≡0. Calculate the required area.

This is the area I would like to calculate

I tried to calculate like this but it does not get me the right answer. Any help?

g = @(x, y) 1./((2+1.1.*x.^2+1.1.*y.^2).^(1/2));

a=0.0;

b=4.40;

c=0.0;

d=4.20;

s1= 1./((2+1.1.*(0.0).^2+1.1.*(0.0).^2).^(1/2));

s2= 1./((2+1.1.*(4.40).^2+1.1.*(4.20).^2).^(1/2));

s3= 1./((2+1.1.*(0.0).^2+1.1.*(4.20).^2).^(1/2));

s4= 1./((2+1.1.*(4.40).^2+1.1.*(0.0).^2).^(1/2));

roof = integral2(g,a,b,c,d);

wall1 = integral2(g,a,b,0,s1);

wall2 = integral2(g,a,b,0,s2);

wall3 = integral2(g,c,d,0,s3);

wall4 = integral2(g,c,d,0,s4);

area = roof + wall1 + wall2 + wall3 + wall4;

##### 1 Comment

Riccardo Scorretti
on 30 Apr 2022

### Accepted Answer

Riccardo Scorretti
on 30 Apr 2022

Edited: Riccardo Scorretti
on 30 Apr 2022

Let's start by checking the function g = profile of your building:

g = @(x, y) 1./((2+1.1.*x.^2+1.1.*y.^2).^(1/2));

a=0.0;

b=4.40;

c=0.0;

d=4.20;

[x, y] = meshgrid(a:0.4:b, c:0.4:d);

figure ; mesh(x, y, g(x,y)) ; axis image ; grid on ;

axis([a, b, c, d, 0 inf]);

camproj perspective ; box on

xlabel('x') ; ylabel('y');

For the 4 walls, the computation boils up into a simple 1D integral:

wall1 = integral(@(x) g(x,c), a, b)

wall3 = integral(@(x) g(x,d), a, b)

and

wall2 = integral(@(y) g(a,y), c, d)

wall4 = integral(@(y) g(b,y), c, d)

The case of the roof is quite different: it is a 2D integral of the surface given defined by the map , that is:

Assuming that g is given as a numerical function (and not analytically), the derivatives must be computed numerically as well, for instance by finite difference. This will introduce some numerical error, which will be quite negligible if the surface is smooth enough:

dx = 1.0E-5 ; dy = dx;

gx = @(x,y) (g(x+dx,y) - g(x-dx,y))/(2*dx);

gy = @(x,y) (g(x,y+dy) - g(x,y-dy))/(2*dy);

roof = integral2(@(x,y) sqrt(1 + gx(x,y).^2 + gy(x,y).^2), a, b, c, d)

The bottom line is:

tot = wall1 + wall2 + wall3 + wall4 + roof

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