Graphic Interpretation for Æ©f(x)Îx

[Jhenrique]

Graphic Interpretation for Ʃf(x)Δx

Hello!

I was trying to understand what means:
\sum_{x_0}^{x_1}f(x)\Delta x
(when Δx = 1 and x ∈ Z, ie, a "discrete integration", topic very comun in discrete calculus).

I computed the result so:
\sum_{1}^{4}x^2\Delta x=\left [\frac{1}{3}x^3-\frac{1}{2}x^2+\frac{1}{6}x \right ]_{1}^{4}=F(4)-F(1)=14
and I sketched the graphic:

However, the Maple computes the result as:
\sum_{1}^{4}x^2\Delta x=>\sum_{1}^{4}x^2\cdot 1=>\sum_{1}^{4}x^2=30
Given a different result of calculated for me. Why?

 

[AlephZero]

Maple is calculating the sum following the conventional rules of notation for $\sum_1^4$, i.e. the sum is 1 + 4 + 9 + 16 = 30.

In discrete calculus you are only summing 3 items, not 4. That is why you got 1 + 4 + 9 = 14.

Maybe there is an option in Maple to use discrete calculus notation, but I don't know about that.

 

[HallsofIvy]

Hello!

I was trying to understand what means:
\sum_{x_0}^{x_1}f(x)\Delta x
(when Δx = 1 and x ∈ Z, ie, a "discrete integration", topic very comun in discrete calculus).

This is not a very good notation! It says you are summing from x_0 to x_1 but does NOT say what step you are using- what \Delt x[/itex is.<br /> <br /> <blockquote data-attributes="" data-quote="" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> I computed the result so:<br /> \sum_{1}^{4}x^2\Delta x=\left [\frac{1}{3}x^3-\frac{1}{2}x^2+\frac{1}{6}x \right ]_{1}^{4}=F(4)-F(1)=14 </div> </div> </blockquote> You appear to be assuming that x_0= 1 and x_1= 4[/itex[ are the only values used- that is, that \Delta x= 4- 1= 3. But this is a &amp;lt;b&amp;gt;sum&amp;lt;/b&amp;gt;, not a difference. It is (1)^2(3)+ (2)^2+ 3= 3+ 12= 15.&amp;lt;br /&amp;gt; &amp;lt;br /&amp;gt; &amp;lt;blockquote data-attributes=&amp;quot;&amp;quot; data-quote=&amp;quot;&amp;quot; data-source=&amp;quot;&amp;quot; class=&amp;quot;bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch&amp;quot;&amp;gt; &amp;lt;div class=&amp;quot;bbCodeBlock-content&amp;quot;&amp;gt; &amp;lt;div class=&amp;quot;bbCodeBlock-expandContent js-expandContent &amp;quot;&amp;gt; and I sketched the graphic:&amp;lt;br /&amp;gt; &amp;lt;div class=&amp;quot;bbImageWrapper js-lbImage&amp;quot; title=&amp;quot;imagem.jpg&amp;quot; data-src=&amp;quot;https://www.physicsforums.com/attachments/imagem-jpg.166287/&amp;quot; data-lb-sidebar-href=&amp;quot;&amp;quot; data-lb-caption-extra-html=&amp;quot;&amp;quot; data-single-image=&amp;quot;1&amp;quot;&amp;gt; &amp;lt;img src=&amp;quot;https://www.physicsforums.com/attachments/imagem-jpg.166287/&amp;quot; data-url=&amp;quot;&amp;quot; class=&amp;quot;bbImage&amp;quot; data-zoom-target=&amp;quot;1&amp;quot; style=&amp;quot;&amp;quot; alt=&amp;quot;imagem.jpg&amp;quot; title=&amp;quot;imagem.jpg&amp;quot; width=&amp;quot;112&amp;quot; height=&amp;quot;180&amp;quot; loading=&amp;quot;lazy&amp;quot; decoding=&amp;quot;async&amp;quot; /&amp;gt; &amp;lt;/div&amp;gt; &amp;lt;/div&amp;gt; &amp;lt;/div&amp;gt; &amp;lt;/blockquote&amp;gt; That graphic shows you using \Delta x= 1, so that x takes on values of 1, 2, and 3:&amp;lt;br /&amp;gt; 1^2(1)+ 2^2(1)+ 3^2(1)= 1+ 4+ 9= 14&amp;lt;br /&amp;gt; &amp;lt;br /&amp;gt; However, the Maple computes the result as:&amp;lt;br /&amp;gt; \sum_{1}^{4}x^2\Delta x=&amp;amp;amp;gt;\sum_{1}^{4}x^2\cdot 1=&amp;amp;amp;gt;\sum_{1}^{4}x^2=30&amp;lt;br /&amp;gt; Given a different result of calculated for me. Why?[/QUOTE]&amp;lt;br /&amp;gt; That sum would NOT approximate the integral from 1 to 4 because it includes a &amp;amp;quot;rectangle&amp;amp;quot; between x= 4 and x= 5: 1+ 4+ 9+ 16= 30.