Tutorial Video
Integration Revision
Let's plot a curve as a a series of bars and a line.
y1=2<span style="color: #000000;">*</span>x1<span style="color: #ff00ff;">;</span>
<span style="color: #0000ff;">figure(</span>1<span style="color: #0000ff;">)</span><span style="color: #ff00ff;">;</span>
<span style="color: #0000ff;">bar(</span>x1<span style="color: #808080;">,</span>y1<span style="color: #0000ff;">)</span><span style="color: #ff00ff;">;</span>
<span style="color: #0000ff;">hold(</span><span style="color: #800080;">'</span><span style="color: #ff9900;">on</span><span style="color: #800080;">'</span><span style="color: #0000ff;">)</span><span style="color: #ff00ff;">;</span>
<span style="color: #0000ff;">plot(</span>x1<span style="color: #808080;">,</span>y1<span style="color: #808080;">,</span><span style="color: #800080;">'</span><span style="color: #ff9900;">LineWidth</span><span style="color: #800080;">'</span><span style="color: #808080;">,</span>3<span style="color: #0000ff;">)</span><span style="color: #ff00ff;">;</span>
<span style="color: #0000ff;">hold(</span><span style="color: #800080;">'</span><span style="color: #ff9900;">off</span><span style="color: #800080;">'</span><span style="color: #0000ff;">)</span><span style="color: #ff00ff;">;</span>
<span style="color: #0000ff;">grid(</span><span style="color: #800080;">'</span><span style="color: #ff9900;">minor</span><span style="color: #800080;">'</span><span style="color: #0000ff;">)</span><span style="color: #ff00ff;">;</span>
<span style="color: #0000ff;">xlabel(</span><span style="color: #800080;">'</span><span style="color: #ff9900;">x</span><span style="color: #800080;">'</span><span style="color: #0000ff;">)</span><span style="color: #ff00ff;">;</span>
<span style="color: #0000ff;">ylabel(</span><span style="color: #800080;">'</span><span style="color: #ff9900;">y</span><span style="color: #800080;">'</span><span style="color: #0000ff;">)</span><span style="color: #ff00ff;">;</span>
ax=<span style="color: #0000ff;">gca</span><span style="color: #ff00ff;">;</span>
ax<span style="color: #ff0000;">.</span>YAxisLocation=<span style="color: #800080;">'</span><span style="color: #ff9900;">right</span><span style="color: #800080;">'</span><span style="color: #ff00ff;">;</span>
We seen before that integration is the sum of the area under the curve (the sum of the bars). We can plot the area under the curve with respect to
. The first integrated
value is going to be the sum of all the bars at
i.e.
and this is going to correspond to a new value of
because we are shifted over by
, the mid point of the step size in
. The next value is going to be the sum of the bars at
i.e.
and correspond to a new value of
and so on and so forth.
<span style="color: #0000ff;">sum(</span><span style="color: #00ccff;">y1(</span>1<span style="color: #ff0000;">:</span>2<span style="color: #00ccff;">)</span><span style="color: #0000ff;">)</span><span style="color: #ff00ff;">;</span><span style="color: #ff0000;">...</span>
<span style="color: #0000ff;">sum(</span><span style="color: #00ccff;">y1(</span>1<span style="color: #ff0000;">:</span>3<span style="color: #00ccff;">)</span><span style="color: #0000ff;">)</span><span style="color: #ff00ff;">;</span><span style="color: #ff0000;">...</span>
<span style="color: #0000ff;">sum(</span><span style="color: #00ccff;">y1(</span>1<span style="color: #ff0000;">:</span>4<span style="color: #00ccff;">)</span><span style="color: #0000ff;">)</span><span style="color: #ff00ff;">;</span><span style="color: #ff0000;">...</span>
<span style="color: #0000ff;">sum(</span><span style="color: #00ccff;">y1(</span>1<span style="color: #ff0000;">:</span>5<span style="color: #00ccff;">)</span><span style="color: #0000ff;">)</span><span style="color: #ff00ff;">;</span><span style="color: #ff0000;">...</span>
<span style="color: #0000ff;">sum(</span><span style="color: #00ccff;">y1(</span>1<span style="color: #ff0000;">:</span>6<span style="color: #00ccff;">)</span><span style="color: #0000ff;">)</span><span style="color: #ff00ff;">;</span><span style="color: #ff0000;">...</span>
<span style="color: #0000ff;">sum(</span><span style="color: #00ccff;">y1(</span>1<span style="color: #ff0000;">:</span>7<span style="color: #00ccff;">)</span><span style="color: #0000ff;">)</span><span style="color: #ff00ff;">;</span><span style="color: #ff0000;">...</span>
<span style="color: #0000ff;">sum(</span><span style="color: #00ccff;">y1(</span>1<span style="color: #ff0000;">:</span>8<span style="color: #00ccff;">)</span><span style="color: #0000ff;">)</span><span style="color: #ff00ff;">;</span><span style="color: #ff0000;">...</span>
<span style="color: #0000ff;">sum(</span><span style="color: #00ccff;">y1(</span>1<span style="color: #ff0000;">:</span>9<span style="color: #00ccff;">)</span><span style="color: #0000ff;">)</span><span style="color: #ff00ff;">;</span><span style="color: #ff0000;">...</span>
<span style="color: #0000ff;">sum(</span><span style="color: #00ccff;">y1(</span>1<span style="color: #ff0000;">:</span>10<span style="color: #00ccff;">)</span><span style="color: #0000ff;">)</span><span style="color: #ff0000;">]</span><span style="color: #ff00ff;">;</span>
x2=<span style="color: #ff0000;">[</span><span style="color: #ff6600;">[</span>1<span style="color: #ff0000;">:</span>10<span style="color: #ff6600;">]</span>+0.5<span style="color: #ff0000;">]</span>'<span style="color: #ff00ff;">;</span>
<span style="color: #0000ff;">figure(</span>2<span style="color: #0000ff;">)</span><span style="color: #ff00ff;">;</span>
<span style="color: #0000ff;">scatter(</span>x1,y1,100,<span style="color: #800080;">'</span><span style="color: #ff9900;">fill</span><span style="color: #800080;">'</span>,<span style="color: #800080;">'</span><span style="color: #ff9900;">r</span><span style="color: #800080;">'</span><span style="color: #0000ff;">)</span><span style="color: #ff00ff;">;</span>
<span style="color: #0000ff;">hold(</span><span style="color: #800080;">'</span><span style="color: #ff9900;">on</span><span style="color: #800080;">'</span><span style="color: #0000ff;">)</span><span style="color: #ff00ff;">;</span>
<span style="color: #0000ff;">scatter(</span>x2,y2,100,<span style="color: #800080;">'</span><span style="color: #ff9900;">fill</span><span style="color: #800080;">'</span>,<span style="color: #800080;">'</span><span style="color: #ff9900;">b</span><span style="color: #800080;">'</span><span style="color: #0000ff;">)</span><span style="color: #ff00ff;">;</span>
<span style="color: #0000ff;">hold(</span><span style="color: #800080;">'</span><span style="color: #ff9900;">off</span><span style="color: #800080;">'</span><span style="color: #0000ff;">)</span><span style="color: #ff00ff;">;</span>
<span style="color: #0000ff;">grid(</span><span style="color: #800080;">'</span><span style="color: #ff9900;">minor</span><span style="color: #800080;">'</span><span style="color: #0000ff;">)</span><span style="color: #ff00ff;">;</span>
<span style="color: #0000ff;">xlabel(</span><span style="color: #800080;">'</span><span style="color: #ff9900;">x</span><span style="color: #800080;">'</span><span style="color: #0000ff;">)</span><span style="color: #ff00ff;">;</span>
<span style="color: #0000ff;">ylabel(</span><span style="color: #800080;">'</span><span style="color: #ff9900;">y</span><span style="color: #800080;">'</span><span style="color: #0000ff;">)</span><span style="color: #ff00ff;">;</span>
ax=<span style="color: #0000ff;">gca</span><span style="color: #ff00ff;">;</span>
ax<span style="color: #ff0000;">.</span>YAxisLocation=<span style="color: #800080;">'</span><span style="color: #ff9900;">right</span>'<span style="color: #ff00ff;">;</span>
<span style="color: #0000ff;">legend(</span><span style="color: #800080;">'</span><span style="color: #ff9900;">y1=2*x1</span><span style="color: #800080;">'</span>,<span style="color: #800080;">'</span><span style="color: #ff9900;">y2=int(y1,x1)</span><span style="color: #800080;">'</span>,<span style="color: #800080;">'</span><span style="color: #ff9900;">location</span><span style="color: #800080;">'</span>,<span style="color: #800080;">'</span><span style="color: #ff9900;">northwest</span><span style="color: #800080;">'</span><span style="color: #0000ff;">)</span><span style="color: #ff00ff;">;</span>
We can plot
,
and
,
as a scatter plot. Noting that
and
and the plots are correct.
Instead of manually typing in
<span style="color: #0000ff;">sum(</span><span style="color: #00ccff;">y1(</span>1<span style="color: #ff0000;">:</span>2<span style="color: #00ccff;">)</span><span style="color: #0000ff;">)</span><span style="color: #ff00ff;">;</span><span style="color: #ff0000;">...</span>
<span style="color: #0000ff;">sum(</span><span style="color: #00ccff;">y1(</span>1<span style="color: #ff0000;">:</span>3<span style="color: #00ccff;">)</span><span style="color: #0000ff;">)</span><span style="color: #ff00ff;">;</span><span style="color: #ff0000;">...</span>
<span style="color: #0000ff;">sum(</span><span style="color: #00ccff;">y1(</span>1<span style="color: #ff0000;">:</span>4<span style="color: #00ccff;">)</span><span style="color: #0000ff;">)</span><span style="color: #ff00ff;">;</span><span style="color: #ff0000;">...</span>
<span style="color: #0000ff;">sum(</span><span style="color: #00ccff;">y1(</span>1<span style="color: #ff0000;">:</span>5<span style="color: #00ccff;">)</span><span style="color: #0000ff;">)</span><span style="color: #ff00ff;">;</span><span style="color: #ff0000;">...</span>
<span style="color: #0000ff;">sum(</span><span style="color: #00ccff;">y1(</span>1<span style="color: #ff0000;">:</span>6<span style="color: #00ccff;">)</span><span style="color: #0000ff;">)</span><span style="color: #ff00ff;">;</span><span style="color: #ff0000;">...</span>
<span style="color: #0000ff;">sum(</span><span style="color: #00ccff;">y1(</span>1<span style="color: #ff0000;">:</span>7<span style="color: #00ccff;">)</span><span style="color: #0000ff;">)</span><span style="color: #ff00ff;">;</span><span style="color: #ff0000;">...</span>
<span style="color: #0000ff;">sum(</span><span style="color: #00ccff;">y1(</span>1<span style="color: #ff0000;">:</span>8<span style="color: #00ccff;">)</span><span style="color: #0000ff;">)</span><span style="color: #ff00ff;">;</span><span style="color: #ff0000;">...</span>
<span style="color: #0000ff;">sum(</span><span style="color: #00ccff;">y1(</span>1<span style="color: #ff0000;">:</span>9<span style="color: #00ccff;">)</span><span style="color: #0000ff;">)</span><span style="color: #ff00ff;">;</span><span style="color: #ff0000;">...</span>
<span style="color: #0000ff;">sum(</span><span style="color: #00ccff;">y1(</span>1<span style="color: #ff0000;">:</span>10<span style="color: #00ccff;">)</span><span style="color: #0000ff;">)</span><span style="color: #ff0000;">]</span><span style="color: #ff00ff;">;</span>
We can use
Differenciation
What we are now going to do is look at the plot
i.e. a constant with respect to
<span style="color: #ff0000;">[</span>m3<span style="color: #808080;">,</span>n3<span style="color: #ff0000;">]</span>=<span style="color: #0000ff;">size(</span>x3<span style="color: #0000ff;">)</span><span style="color: #ff00ff;">;</span>
y3=<span style="color: #0000ff;">ones(</span>m3,n3<span style="color: #0000ff;">)</span><span style="color: #ff00ff;">;</span>
<span style="color: #0000ff;">figure(</span>3<span style="color: #0000ff;">)</span><span style="color: #ff00ff;">;</span>
<span style="color: #0000ff;">scatter(</span>x3<span style="color: #808080;">,</span>y3<span style="color: #808080;">,</span>100<span style="color: #808080;">,</span><span style="color: #800080;">'</span><span style="color: #ff9900;">fill</span><span style="color: #800080;">'</span>,<span style="color: #800080;">'</span><span style="color: #ff9900;">r</span><span style="color: #800080;">'</span><span style="color: #0000ff;">)</span><span style="color: #ff00ff;">;</span>
<span style="color: #0000ff;">hold(</span><span style="color: #800080;">'</span><span style="color: #ff9900;">on</span><span style="color: #800080;">'</span><span style="color: #0000ff;">)</span><span style="color: #ff00ff;">;</span>
<span style="color: #0000ff;">scatter(</span>x4<span style="color: #808080;">,</span>y4<span style="color: #808080;">,</span>100<span style="color: #808080;">,</span><span style="color: #800080;">'</span><span style="color: #ff9900;">fill</span><span style="color: #800080;">'</span><span style="color: #808080;">,</span><span style="color: #800080;">'</span><span style="color: #ff9900;">b</span><span style="color: #800080;">'</span><span style="color: #0000ff;">)</span><span style="color: #ff00ff;">;</span>
<span style="color: #0000ff;">hold(</span><span style="color: #800080;">'</span><span style="color: #ff9900;">off</span><span style="color: #800080;">'</span><span style="color: #0000ff;">)</span><span style="color: #ff00ff;">;</span>
<span style="color: #0000ff;">grid(</span><span style="color: #800080;">'</span><span style="color: #ff9900;">minor</span><span style="color: #800080;">'</span><span style="color: #0000ff;">)</span><span style="color: #ff00ff;">;</span>
<span style="color: #0000ff;">xlabel(</span><span style="color: #800080;">'</span><span style="color: #ff9900;">x</span><span style="color: #800080;">'</span><span style="color: #0000ff;">)</span><span style="color: #ff00ff;">;</span>
<span style="color: #0000ff;">ylabel(</span><span style="color: #800080;">'</span><span style="color: #ff9900;">y</span><span style="color: #800080;">'</span><span style="color: #0000ff;">)</span><span style="color: #ff00ff;">;</span>
ax=<span style="color: #0000ff;">gca</span><span style="color: #ff00ff;">;</span>
ax<span style="color: #ff0000;">.</span>YAxisLocation=<span style="color: #800080;">'</span><span style="color: #ff9900;">right</span>'<span style="color: #ff00ff;">;</span>
<span style="color: #0000ff;">legend(</span><span style="color: #800080;">'</span><span style="color: #ff9900;">y1=1</span><span style="color: #800080;">'</span><span style="color: #808080;">,</span><span style="color: #800080;">'</span><span style="color: #ff9900;">2=int(y1,x1)</span><span style="color: #800080;">'</span><span style="color: #808080;">,</span><span style="color: #800080;">'</span><span style="color: #ff9900;">location</span><span style="color: #800080;">'</span><span style="color: #808080;">,</span><span style="color: #800080;">'</span><span style="color: #ff9900;">northwest</span><span style="color: #800080;">'</span><span style="color: #0000ff;">)</span><span style="color: #ff00ff;">;</span>
Now instead of looking at the area under the line, we are going to look at the difference between the nearest
values i.e. the difference between a bar with the bar on its left. For instance the tenth bar at
and the 9th bar at
is going to correspond to a new value of
that is 0.5 less than 9 i.e.
. The next value is going to be the difference of the points at
and
and correspond to a new value of
that is 0.5 less than i.e.
and so on and so forth.
We will plot this as a scatter plot on the same chart.
<span style="color: #ff6600;">[</span><span style="color: #0000ff;">y3(</span>2<span style="color: #0000ff;">)</span>-<span style="color: #0000ff;">y3(</span>1<span style="color: #0000ff;">)</span><span style="color: #ff6600;">]</span><span style="color: #ff00ff;">;</span><span style="color: #ff0000;">...</span>
<span style="color: #ff6600;">[</span><span style="color: #0000ff;">y3(</span>3<span style="color: #0000ff;">)</span>-<span style="color: #0000ff;">y3(</span>2<span style="color: #0000ff;">)</span><span style="color: #ff6600;">]</span><span style="color: #ff00ff;">;</span><span style="color: #ff0000;">...</span>
<span style="color: #ff6600;">[</span><span style="color: #0000ff;">y3(</span>4<span style="color: #0000ff;">)</span>-<span style="color: #0000ff;">y3(<span style="color: #000000;">3</span></span><span style="color: #0000ff;">)</span><span style="color: #ff6600;">]</span><span style="color: #ff00ff;">;</span><span style="color: #ff0000;">...</span>
<span style="color: #ff6600;">[</span><span style="color: #0000ff;">y3(</span>5<span style="color: #0000ff;">)</span>-<span style="color: #0000ff;">y3(</span>4<span style="color: #0000ff;">)</span><span style="color: #ff6600;">]</span><span style="color: #ff00ff;">;</span><span style="color: #ff0000;">...</span>
<span style="color: #ff6600;">[</span><span style="color: #0000ff;">y3(</span>6<span style="color: #0000ff;">)</span>-<span style="color: #0000ff;">y3(</span>5<span style="color: #0000ff;">)</span><span style="color: #ff6600;">]</span><span style="color: #ff00ff;">;</span><span style="color: #ff0000;">...</span>
<span style="color: #ff6600;">[</span><span style="color: #0000ff;">y3(</span>7<span style="color: #0000ff;">)</span>-<span style="color: #0000ff;">y3(</span>6<span style="color: #0000ff;">)</span><span style="color: #ff6600;">]</span><span style="color: #ff00ff;">;</span><span style="color: #ff0000;">...</span>
<span style="color: #ff6600;">[</span><span style="color: #0000ff;">y3(</span>8<span style="color: #0000ff;">)</span>-<span style="color: #0000ff;">y3(</span>7<span style="color: #0000ff;">)</span><span style="color: #ff6600;">]</span><span style="color: #ff00ff;">;</span><span style="color: #ff0000;">...</span>
<span style="color: #ff6600;">[</span><span style="color: #0000ff;">y3(</span>9<span style="color: #0000ff;">)</span>-<span style="color: #0000ff;">y3(</span>8<span style="color: #0000ff;">)</span><span style="color: #ff6600;">]</span><span style="color: #ff00ff;">;</span><span style="color: #ff0000;">...</span>
<span style="color: #ff6600;">[</span><span style="color: #0000ff;">y3(</span>10<span style="color: #0000ff;">)</span>-<span style="color: #0000ff;">y3(</span>9<span style="color: #0000ff;">)</span><span style="color: #ff6600;">]</span><span style="color: #ff0000;">]</span><span style="color: #ff00ff;">;</span>
x4=<span style="color: #ff0000;">[</span><span style="color: #ff6600;">[</span>1<span style="color: #ff0000;">:</span>10<span style="color: #ff6600;">]</span>-0.5<span style="color: #ff0000;">]</span><span style="color: #800080;">'</span><span style="color: #ff00ff;">;</span>
<span style="color: #0000ff;">figure(</span>4<span style="color: #0000ff;">)</span><span style="color: #ff00ff;">;</span>
<span style="color: #0000ff;">scatter(</span>x3<span style="color: #808080;">,</span>y3<span style="color: #808080;">,</span>100<span style="color: #808080;">,</span><span style="color: #800080;">'</span><span style="color: #ff9900;">fill</span><span style="color: #800080;">'</span>,<span style="color: #800080;">'</span><span style="color: #ff9900;">r</span><span style="color: #800080;">'</span><span style="color: #0000ff;">)</span><span style="color: #ff00ff;">;</span>
<span style="color: #0000ff;">hold(</span><span style="color: #800080;">'</span><span style="color: #ff9900;">on</span><span style="color: #800080;">'</span><span style="color: #0000ff;">)</span><span style="color: #ff00ff;">;</span>
<span style="color: #0000ff;">scatter(</span>x4<span style="color: #808080;">,</span>y4<span style="color: #808080;">,</span>100<span style="color: #808080;">,</span><span style="color: #800080;">'</span><span style="color: #ff9900;">fill</span><span style="color: #800080;">'</span>,<span style="color: #800080;">'</span><span style="color: #ff9900;">b</span><span style="color: #800080;">'</span><span style="color: #0000ff;">)</span><span style="color: #ff00ff;">;</span>
<span style="color: #0000ff;">hold(</span><span style="color: #800080;">'</span><span style="color: #ff9900;">off</span><span style="color: #800080;">'</span><span style="color: #0000ff;">)</span><span style="color: #ff00ff;">;</span>
<span style="color: #0000ff;">grid(</span><span style="color: #800080;">'</span><span style="color: #ff9900;">minor</span><span style="color: #800080;">'</span><span style="color: #0000ff;">)</span><span style="color: #ff00ff;">;</span>
<span style="color: #0000ff;">xlabel(</span><span style="color: #800080;">'</span><span style="color: #ff9900;">x</span><span style="color: #800080;">'</span><span style="color: #0000ff;">)</span><span style="color: #ff00ff;">;</span>
<span style="color: #0000ff;">ylabel(</span><span style="color: #800080;">'</span><span style="color: #ff9900;">y</span><span style="color: #800080;">'</span><span style="color: #0000ff;">)</span><span style="color: #ff00ff;">;</span>
ax=<span style="color: #0000ff;">gca</span><span style="color: #ff00ff;">;</span>
ax<span style="color: #ff0000;">.</span>YAxisLocation=<span style="color: #800080;">'</span><span style="color: #ff9900;">right</span><span style="color: #800080;">'</span><span style="color: #ff00ff;">;</span>
<span style="color: #0000ff;">legend(</span><span style="color: #800080;">'</span><span style="color: #ff9900;">y3=1</span><span style="color: #800080;">'</span><span style="color: #808080;">,</span><span style="color: #800080;">'</span><span style="color: #ff9900;">y4=diff(y3,x3)</span><span style="color: #800080;">'</span><span style="color: #808080;">,</span><span style="color: #800080;">'</span><span style="color: #ff9900;">location</span><span style="color: #800080;">'</span><span style="color: #808080;">,</span><span style="color: #800080;">'</span><span style="color: #ff9900;">NorthWest</span><span style="color: #800080;">'</span><span style="color: #0000ff;">)</span><span style="color: #ff00ff;">;</span>
In this case since
is a constant, the difference in the
values
is zero across the board as expected.
Instead of manually typing in:
<span style="color: #ff6600;">[</span><span style="color: #0000ff;">y3(</span>2<span style="color: #0000ff;">)</span>-<span style="color: #0000ff;">y3(</span>1<span style="color: #0000ff;">)</span><span style="color: #ff6600;">]</span><span style="color: #ff00ff;">;</span><span style="color: #ff0000;">...</span>
<span style="color: #ff6600;">[</span><span style="color: #0000ff;">y3(</span>3<span style="color: #0000ff;">)</span>-<span style="color: #0000ff;">y3(</span>2<span style="color: #0000ff;">)</span><span style="color: #ff6600;">]</span><span style="color: #ff00ff;">;</span><span style="color: #ff0000;">...</span>
<span style="color: #ff6600;">[</span><span style="color: #0000ff;">y3(</span>4<span style="color: #0000ff;">)</span>-<span style="color: #0000ff;">y3(<span style="color: #000000;">3</span></span><span style="color: #0000ff;">)</span><span style="color: #ff6600;">]</span><span style="color: #ff00ff;">;</span><span style="color: #ff0000;">...</span>
<span style="color: #ff6600;">[</span><span style="color: #0000ff;">y3(</span>5<span style="color: #0000ff;">)</span>-<span style="color: #0000ff;">y3(</span>4<span style="color: #0000ff;">)</span><span style="color: #ff6600;">]</span><span style="color: #ff00ff;">;</span><span style="color: #ff0000;">...</span>
<span style="color: #ff6600;">[</span><span style="color: #0000ff;">y3(</span>6<span style="color: #0000ff;">)</span>-<span style="color: #0000ff;">y3(</span>5<span style="color: #0000ff;">)</span><span style="color: #ff6600;">]</span><span style="color: #ff00ff;">;</span><span style="color: #ff0000;">...</span>
<span style="color: #ff6600;">[</span><span style="color: #0000ff;">y3(</span>7<span style="color: #0000ff;">)</span>-<span style="color: #0000ff;">y3(</span>6<span style="color: #0000ff;">)</span><span style="color: #ff6600;">]</span><span style="color: #ff00ff;">;</span><span style="color: #ff0000;">...</span>
<span style="color: #ff6600;">[</span><span style="color: #0000ff;">y3(</span>8<span style="color: #0000ff;">)</span>-<span style="color: #0000ff;">y3(</span>7<span style="color: #0000ff;">)</span><span style="color: #ff6600;">]</span><span style="color: #ff00ff;">;</span><span style="color: #ff0000;">...</span>
<span style="color: #ff6600;">[</span><span style="color: #0000ff;">y3(</span>9<span style="color: #0000ff;">)</span>-<span style="color: #0000ff;">y3(</span>8<span style="color: #0000ff;">)</span><span style="color: #ff6600;">]</span><span style="color: #ff00ff;">;</span><span style="color: #ff0000;">...</span>
<span style="color: #ff6600;">[</span><span style="color: #0000ff;">y3(</span>10<span style="color: #0000ff;">)</span>-<span style="color: #0000ff;">y3(</span>9<span style="color: #0000ff;">)</span><span style="color: #ff6600;">]</span><span style="color: #ff0000;">]</span><span style="color: #ff00ff;">;</span>
We can use
x4=<span style="color: #ff0000;">[</span><span style="color: #ff9900;">[</span>1<span style="color: #ff0000;">:</span>10<span style="color: #ff9900;">]</span>-0.5<span style="color: #ff0000;">]</span>'<span style="color: #ff00ff;">;</span>
A value for
was not selected and is the reason for
opposed to its actual value of
. As this is an outlier it is better to ignore this first value and instead look at:
x4=<span style="color: #ff0000;">[</span><span style="color: #ff9900;">[</span>2<span style="color: #ff0000;">:</span>10<span style="color: #ff9900;">]</span>-0.5<span style="color: #ff0000;">]</span>'<span style="color: #ff00ff;">;</span>
Let's instead change the
(constant) to
. Now the bar chart from right to left looks like a set of stairs and as we climb down the step size remains constant:
As a result
is a straight line and is independent of
.
Okay so let's now change
. Here the bar chart from right to left stepping down we see the first step is the larges, the the second step, then the third and it is a much shallower step down on the left hand side:
Actually we can see that every value of
is equal to
i.e.
Integration vs Differentiation
The integration of
with respect to
gives
and the differentiation of
with respect to
gives
i.e. as you can clearly see they are the inverse process of one another.
We can think of integration as adding the sum of all the bars as we move from left to right – climbing the stairs i.e. starting at the origin and our first plot point being
).
And we can think of differentiation as being the difference in the bar height as we step down from the right to left. The first step from the right being
.
Of course as we seen with integration the above is only an approximation as we used finite relatively large step sizes.
Uncertainty in Results
In the above we had the bin width equaling to the finite bin width of 1. For integration and differentiation we use an infinitely small bin width. Let's constrict our bin width
Rules of Integration and Differenciation
Taking
For integration we elevate the original power by 1 and then divide by the new power.
For differentiation we multiply by the original power and then devalue the power by 1:
Symbolic
We can set x to be a symbol and write down our starting equations with respect to x:
z1=1
z2=x
z3=x^2
z4=x^3
Now we can differentiate the functions with respect to x:
dz2=<span style="color: #0000ff;">diff(</span>z2<span style="color: #808080;">,</span>x<span style="color: #0000ff;">)</span>
dz3=<span style="color: #0000ff;">diff(</span>z3<span style="color: #808080;">,</span>x<span style="color: #0000ff;">)</span>
dz4=<span style="color: #0000ff;">diff(</span>z4<span style="color: #808080;">,</span>x<span style="color: #0000ff;">)</span>
Now we can integrate the functions with respect to x:
iz2=<span style="color: #0000ff;">int(</span>z2<span style="color: #808080;">,</span>x<span style="color: #0000ff;">)</span>
iz3=<span style="color: #0000ff;">int(</span>z3<span style="color: #808080;">,</span>x<span style="color: #0000ff;">)</span>
iz4=<span style="color: #0000ff;">int(</span>z4<span style="color: #808080;">,</span>x<span style="color: #0000ff;">)</span>