# set terminal pngcairo  background "#ffffff" enhanced font "arial,10" fontscale 1.0 size 600, 400 
# set output 'bivariat.3.png'
set key inside right bottom vertical Right noreverse enhanced autotitles nobox
set samples 50, 50
set style data lines
set title "approximate the integral of functions (upper and lower limits)" 
integral_f(x) = (x>0)?int1a(x,x/ceil(x/delta)):-int1b(x,-x/ceil(-x/delta))
int1a(x,d) = (x<=d*.1) ? 0 : (int1a(x-d,d)+(f(x-d)+4*f(x-d*.5)+f(x))*d/6.)
int1b(x,d) = (x>=-d*.1) ? 0 : (int1b(x+d,d)+(f(x+d)+4*f(x+d*.5)+f(x))*d/6.)
f(x)=(x-2)**2-20
integral2_f(x,y) = (x<y)?int2(x,y,(y-x)/ceil((y-x)/delta)):                         -int2(y,x,(x-y)/ceil((x-y)/delta))
int2(x,y,d) = (x>y-d*.5) ? 0 : (int2(x+d,y,d) + (f(x)+4*f(x+d*.5)+f(x+d))*d/6.)
delta = 0.2
GPFUN_integral_f = "integral_f(x) = (x>0)?int1a(x,x/ceil(x/delta)):-int1b(x,-x/ceil(-x/delta))"
GPFUN_int1a = "int1a(x,d) = (x<=d*.1) ? 0 : (int1a(x-d,d)+(f(x-d)+4*f(x-d*.5)+f(x))*d/6.)"
GPFUN_int1b = "int1b(x,d) = (x>=-d*.1) ? 0 : (int1b(x+d,d)+(f(x+d)+4*f(x+d*.5)+f(x))*d/6.)"
GPFUN_integral2_f = "integral2_f(x,y) = (x<y)?int2(x,y,(y-x)/ceil((y-x)/delta)):                         -int2(y,x,(x-y)/ceil((x-y)/delta))"
GPFUN_int2 = "int2(x,y,d) = (x>y-d*.5) ? 0 : (int2(x+d,y,d) + (f(x)+4*f(x+d*.5)+f(x+d))*d/6.)"
GPFUN_f = "f(x)=(x-2)**2-20"
plot [-10:10] f(x) title "f(x)=(x-2)**2-20", integral2_f(-5,x)