
Sum Property 
Scalar Multiple Property

Addition Property 

$\int_a^b P(x) + Q(x) \ dx = \int_a^b P(x)\ dx + \int_a^b Q(x) \ dx $ 
$\int_a^b \alpha P(x)\ dx = \alpha \int_a^b P(x)\ dx$

$\int_a^b P(x)\ dx = \int_a^c P(x)\ dx + \int_c^b P(x) \ dx $ 
AreaGraph Interpretation

The area interpretation visualizes the graphs of the functions,
$P$and $Q$, with the graph of the function $P+Q$ on the same interval $[a,b]$ and the property
is understood by recognizing the region determined by $P+Q$ as
combining the regions determined by $P$ and $Q$ with the areas (definite
integrals) being added.

The
area interpretation visualizes the graphs of the functions,
$P$ and $\alpha P$, with the graph of the function $\alpha P$ on the
same interval $[a,b]$ and the property
is understood by recognizing the region determined by $\alpha P$ as
scaling the region determined by $P$ the area (definite
integral) being multiplied by $\alpha$. 
The area interpretation visualizes the graph of the function,
$P$, on the intervals $[a,b]$,$[a,c]$, and $[c,b]$ and the property
is understood by recognizing the regions determined the region
determined by $[a,b]$ as combining the regions determined by $[a,c]$ and
$[c,b]$ with the areas (definite integrals) being added. 
MotionMapping Diagram Interpretation

The mapping diagram visualizes the integrals as the net change in
position for an object moving over the time interval $[a,b]$ with
velocities of $P,Q$, and $P+Q$. The result for the object moving with
the velocity $P+Q$ (that adds the separate velocities) will add the net changes
for objects traveling with the separate velocities.

The mapping diagram visualizes the integrals as the net change in
position for an object moving over the time interval $[a,b]$ with
velocities of $P$ and $\alpha P$. The result for the object moving with
the velocity $\alpha P$ (that scales the original velocity) will multiply the net change
for object traveling with the velocity $P$ by $\alpha$. 
The mapping diagram visualizes the integrals as the net change in
position for an object moving over the time intervals $[a,b]$,$[a,c]$, and $[c,b]$ with
velocity of $P$. The result for the object moving with
the velocity $P$ over the interval $[a,b]$ will add the net changes
for object traveling over the separate intervals.
