Question 1029380
{{{sum( ((3i/6)^3-6(3i/6))*(3/6),matrix(1,3,"","",i=1),matrix(1,4,"","","",6))}}}
<pre>
represents the sum of the 6 rectangles below:

{{{drawing(200,708&1/3,-.9,3.9,-7,10,graph(200,708&1/3,-.9,3.9,-7,10,x^3-6x),
line(3,0,3,9),line(2,0,2,-4),line(2,0,2,.625),
line(1.5,-5.625,1.5,0),line(1,0,1,-5.625),line(.5,-5,.5,0),line(2,.625,2.5,.625),
line(2.5,9,3,9),line(2.5,0,2.5,9),
line(0,0,0,-2.875),line(0,-2.875,.5,-2.875),line(.5,-5,1,-5),line(1,-5.625,1.5,-5.625),line(1.5,-4,2,-4) )}}}

Now we simplify the sum:

{{{sum( ((3i/6)^3-6(3i/6))*(3/6),matrix(1,3,"","",i=1),matrix(1,4,"","","",6))}}}{{{""=""}}}

{{{sum( ((i/2)^3-6(i/2))*(1/2),matrix(1,3,"","",i=1),matrix(1,4,"","","",6))}}}{{{""=""}}}

{{{sum( (i^3/8-3i)*(1/2),matrix(1,3,"","",i=1),matrix(1,4,"","","",6))}}}{{{""=""}}}

{{{sum( (i^3/8-24i/8)*(1/2),matrix(1,3,"","",i=1),matrix(1,4,"","","",6))}}}{{{""=""}}}

{{{sum(expr(((i^3-24i)/8)*(1/2)),matrix(1,3,"","",i=1),matrix(1,4,"","","",6))}}}{{{""=""}}}

{{{sum( ((i^3-24i)/16),matrix(1,3,"","",i=1),matrix(1,4,"","","",6))}}}{{{""=""}}}

{{{expr(1/16)sum( (i^3-24i),matrix(1,2,"",i=1),matrix(1,3,"","",6))}}}{{{""=""}}}

{{{expr(1/16)sum( (i(i^2-24)^""),matrix(1,2,"",i=1),matrix(1,3,"","",6))}}}{{{""=""}}}

{{{expr(1/16)*(1(1^2-24)+2(2^2-24)+3(3^2-24)+4(4^2-24)+5(5^2-24)+6(6^2-24)^"")}}}{{{""=""}}}

{{{expr(1/16)*(1(1-24)+2(4-24)+3(9-24)+4(16-24)+5(25-24)+6(36-24)^"")}}}{{{""=""}}}

{{{expr(1/16)*(1(-23)+2(-20)+3(-15)+4(-8)+5(1)+6(12)^"")}}}{{{""=""}}}

{{{expr(1/16)*(-23-40-45-32+5+72)}}}{{{""=""}}}

{{{expr(1/16)*(-63)}}}{{{""=""}}}

{{{-3.9375}}}

Three decimal places

{{{-3.938}}}

Edwin</pre>