# set terminal pngcairo  transparent enhanced font "arial,10" fontscale 1.0 size 600, 400 
# set output 'bivariat.3.png'
set bar 1.000000 front
set style circle radius graph 0.02, first 0.00000, 0.00000 
set style ellipse size graph 0.05, 0.03, first 0.00000 angle 0 units xy
set key inside right bottom vertical Right noreverse enhanced autotitle nobox
set style textbox transparent margins  1.0,  1.0 border
unset logscale
set samples 50, 50
set style data lines
unset paxis 1 tics
unset paxis 2 tics
unset paxis 3 tics
unset paxis 4 tics
unset paxis 5 tics
unset paxis 6 tics
unset paxis 7 tics
set title "approximate the integral of functions (upper and lower limits)" 
set paxis 1 range [ * : * ] noreverse nowriteback
set paxis 2 range [ * : * ] noreverse nowriteback
set paxis 3 range [ * : * ] noreverse nowriteback
set paxis 4 range [ * : * ] noreverse nowriteback
set paxis 5 range [ * : * ] noreverse nowriteback
set paxis 6 range [ * : * ] noreverse nowriteback
set paxis 7 range [ * : * ] noreverse nowriteback
set colorbox vertical origin screen 0.9, 0.2, 0 size screen 0.05, 0.6, 0 front  noinvert bdefault
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)