Let's say for example we have a constant function f(x)= c
Then, its indefinite integral is = cx + g
What is that other constant "g" doing there??? where did it come from and what does it mean???
Geometrically, I expected that the integral would be equal to the area of a rectangle with height "c" and base "x" ... so "cx" would be more than enough to describe the area under the curve (or line). But "cx+g" is the area of a rectangle plus something...what "something"??
Even more strange , when solving Ordinary Differential Equations at the uiniversity we were solving the radioactive decay . After doing a dimensional analysis, it turned out that the constant "g" after integration has units that don´t even belong to the original plane of variables, it gives something that is in a different dimension!:
The equation for the rate of decay is:
dN/dt = -λN
N is the number of atoms, t is the time, λi is a constant of decay with frequency units (1/s)
1. So to solve that differential equation we start by rearranging the equation:
1/N*dN = -λdt
2. then integrating each side:
lnN = -λt + g <-------here´s our magickal constant.
3. So to know what g is ,we know that At time 0 : t = 0 and N = N0 (the initial number of atoms
so at t=0 , lnN0 = g
So far, g is a dimensionless constant but instead of being related to frequency or time , (which should be, because it came from integrating frequency against time)...now it is related to the Number of atoms!!!! why?!?!?!?!
if I take exp(g) it will give me the number of atoms at time 0, i.e.N0, How on earth is it possible that by integrating frequency (1/s) and time (s) I get a number related to the of atoms!!! and back to the original question where did that number even come from, when "-λt" alone describes the area under the curve??
Then, its indefinite integral is = cx + g
What is that other constant "g" doing there??? where did it come from and what does it mean???
Geometrically, I expected that the integral would be equal to the area of a rectangle with height "c" and base "x" ... so "cx" would be more than enough to describe the area under the curve (or line). But "cx+g" is the area of a rectangle plus something...what "something"??
Even more strange , when solving Ordinary Differential Equations at the uiniversity we were solving the radioactive decay . After doing a dimensional analysis, it turned out that the constant "g" after integration has units that don´t even belong to the original plane of variables, it gives something that is in a different dimension!:
The equation for the rate of decay is:
dN/dt = -λN
N is the number of atoms, t is the time, λi is a constant of decay with frequency units (1/s)
1. So to solve that differential equation we start by rearranging the equation:
1/N*dN = -λdt
2. then integrating each side:
lnN = -λt + g <-------here´s our magickal constant.
3. So to know what g is ,we know that At time 0 : t = 0 and N = N0 (the initial number of atoms
so at t=0 , lnN0 = g
So far, g is a dimensionless constant but instead of being related to frequency or time , (which should be, because it came from integrating frequency against time)...now it is related to the Number of atoms!!!! why?!?!?!?!
if I take exp(g) it will give me the number of atoms at time 0, i.e.N0, How on earth is it possible that by integrating frequency (1/s) and time (s) I get a number related to the of atoms!!! and back to the original question where did that number even come from, when "-λt" alone describes the area under the curve??