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.
9
Y la primitiva F de f; And the "primitive" (antiderivative) of f is F:
10
Expression 11: "F" left parenthesis, "x" , right parenthesis equals left brace, "B" greater than "x" greater than "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 , 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" , right parenthesis , right braceFx=B>x>A:modx−a,C·fx−modx−a,C+floorx−aC−1∑n=0C·fa+Cn
11
I added the b> to the x>a expression for F, so it draws the area-so-far as we increase b.
12
Expression 13: "q" left parenthesis, "x" , right parenthesis equals StartFraction, left parenthesis, "F" left parenthesis, "x" plus "C" , right parenthesis minus "F" left parenthesis, "x" , right parenthesis , right parenthesis Over "C" , EndFractionqx=Fx+C−FxC
13
Based on "Integral de cualquier f(x)" by Solin; modified to work for b<a, and added the difference-quotient-of-F graph.
14
Expression 15: "A" equals min left parenthesis, "a" , "b" , right parenthesisA=mina,b
equals=
00
15
Expression 16: "B" equals max left parenthesis, "a" , "b" , right parenthesisB=maxa,b
equals=
4.74.7
16
Expression 17: "C" equals signum left parenthesis, "b" minus "a" , right parenthesis times "c"C=signb−a·c