- #1
yungman
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- 291
This is a copy of the book:
[PLAIN]http://i38.tinypic.com/20faqnc.jpg[/PLAIN]
I know the derivation part, I just want to see whether I understand why the -ve sign of ##-\frac {\partial f}{\partial y}dA## in a more common sense way.
From looking at the graph for type I region, ##g_2(x)## is above ##g_1(x)## referenced to y axis. So the integral has to be ##g_2(x)-g_1(x)##. BUT the orientation of curve of ##g_2(x)## is from b to a. So if we want to integrate from a to b, we need to put a -ve sign.
From the type II region, ##h_2(x)## is above ##h_1(x)## referenced to x axis. So the integral has to be ##h_2(x)-h_1(x)##. The orientation of curve of ##h_2(x)## is from c to d. So if we want to integrate from c to d, it would be +ve sign.
Am I getting it right?
Bottom line is the sign depends on the direction of the higher value function of the two ( ie. ##g_2(t)≥g_1(t)##). If the direction is from high value to low value, then the sign has to be change to make it from low to high ( ie. ##g_2(t)## oriented from b to a. So sign needed to be change to integrate from a to b).
[PLAIN]http://i38.tinypic.com/20faqnc.jpg[/PLAIN]
I know the derivation part, I just want to see whether I understand why the -ve sign of ##-\frac {\partial f}{\partial y}dA## in a more common sense way.
From looking at the graph for type I region, ##g_2(x)## is above ##g_1(x)## referenced to y axis. So the integral has to be ##g_2(x)-g_1(x)##. BUT the orientation of curve of ##g_2(x)## is from b to a. So if we want to integrate from a to b, we need to put a -ve sign.
From the type II region, ##h_2(x)## is above ##h_1(x)## referenced to x axis. So the integral has to be ##h_2(x)-h_1(x)##. The orientation of curve of ##h_2(x)## is from c to d. So if we want to integrate from c to d, it would be +ve sign.
Am I getting it right?
Bottom line is the sign depends on the direction of the higher value function of the two ( ie. ##g_2(t)≥g_1(t)##). If the direction is from high value to low value, then the sign has to be change to make it from low to high ( ie. ##g_2(t)## oriented from b to a. So sign needed to be change to integrate from a to b).
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