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Sum Property |
Scalar Multiple Property
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Addition Property |
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$\int_a^b P(x) + Q(x) \ dx = \int_a^b P(x)\ dx + \int_a^b Q(x) \ dx $ |
$\int_a^b \alpha P(x)\ dx = \alpha \int_a^b P(x)\ dx$
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$\int_a^b P(x)\ dx = \int_a^c P(x)\ dx + \int_c^b P(x) \ dx $ |
Area-Graph Interpretation
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 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.
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 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. |
Motion-Mapping Diagram Interpretation
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 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.
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 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.
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