El area es la parte azul menos la naranja; the area is the blue part minus the orange. Except, if b<a, then the blue areas (above the x-axis) count as negative, and the orange areas (below the x-axis) count as positive.
13
Y en negro la funcion y=F(x) que cuenta este area desde a hasta x.
14
Y la primitiva F de f; And the "primitive" (antiderivative) of f is F:
15
Expression 16: "F" left parenthesis, "x" , right parenthesis equals left brace, "B" greater than or equal to "x" greater than or equal to "A" : mod left parenthesis, "x" minus "a" , "C" , right parenthesis times "f" left parenthesis, "x" minus mod left parenthesis, "x" minus "a" , "C" , right parenthesis plus "s" times StartFraction, "c" Over 2 , EndFraction , right parenthesis plus Start sum from "n" equals 0 to floor left parenthesis, StartFraction, "x" minus "a" Over "C" , EndFraction minus 1 , right parenthesis , end sum, "C" times "f" left parenthesis, "a" plus "C" "n" plus "s" times StartFraction, "c" Over 2 , EndFraction , right parenthesis , right braceFx=B≥x≥A:modx−a,C·fx−modx−a,C+s·c2+floorx−aC−1∑n=0C·fa+Cn+s·c2
16
I added the b> to the x>a expression for F, so it draws the area-so-far as we increase b.
17
Expression 18: "q" left parenthesis, "x" , right parenthesis equals StartFraction, "F" left parenthesis, "x" plus "C" , right parenthesis minus "F" left parenthesis, "x" , right parenthesis Over "C" , EndFractionqx=Fx+C−FxC
18
"x"x
"F" left parenthesis, "x" , right parenthesisFx
"a" plus "c"a+c
r1c2: 1.4 8 6 7 2 times 10 to the negative 8th power1.48672×10−8
r1c3:
"a" plus 2 "c"a+2c
r2c2: 3.0 4 9 5 8 7 times 10 to the negative 8th power3.049587×10−8
r2c3:
"a" plus 3 "c"a+3c
r3c2: 4.6 9 2 3 3 7 times 10 to the negative 8th power4.692337×10−8
r3c3:
"a" plus 4 "c"a+4c
r4c2: 6.4 1 8 8 8 2 times 10 to the negative 8th power6.418882×10−8
r4c3:
"a" plus 5 "c"a+5c
r5c2: 8.2 3 3 3 1 3 times 10 to the negative 8th power8.233313×10−8
r5c3:
"a" plus 6 "c"a+6c
r6c2: 1.0 1 3 9 9 1 times 10 to the negative 7th power1.013991×10−7
r6c3:
r7c2:
r7c3:
19
Based on "Integral de cualquier f(x)" and "Integrate any f(x) by left, right or middle", by Solin; modified to work for b<a, and added the difference-quotient-of-F graph.
20
Expression 21: "A" equals min left parenthesis, "a" , "b" , right parenthesisA=mina,b
equals=
negative 5−5
21
Expression 22: "B" equals max left parenthesis, "a" , "b" , right parenthesisB=maxa,b
equals=
negative 0.5 1 6−0.516
22
Expression 23: "C" equals signum left parenthesis, "b" minus "a" , right parenthesis times "c"C=signb−a·c
equals=
0.0 10.01
23
Hidden Label: left parenthesis, "a" , 0 , right parenthesisa,0
Label
equals=
left parenthesis, negative 5 , 0 , right parenthesis−5,0
24
Hidden Label: left parenthesis, "b" , 0 , right parenthesisb,0
Label
equals=
left parenthesis, negative 0.5 1 6 , 0 , right parenthesis−0.516,0
