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AakashPandita
- 157
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[tex] ∫x(dx)^2=?[/tex]
and what is the difference between dx^2 and (dx)^2?
and what is the difference between dx^2 and (dx)^2?
jackmell said:Isn't that just short-hand notation for a double integral, i.e.,
[tex]\iint x dx dx[/tex]
pwsnafu said:I don't think you are allowed to have the same variable for multiple integrals. A double integral is integrating with respect to the area element.
I strongly suspect that you have misreadjackmell said:Ok, but the reason I suggested this is because I recall in one of my books on integral equations, the author, when converting from IVP to Volterra Equations, use the syntax such as:
[tex]\int_0^x \int_0^x f(t) dt dt=\int_0^x (x-t) f(t)dt[/tex]
Neither "[itex]dtdt[/itex]" nor "[itex](dx)^2[/itex]" has any meaning.however, to be fair, he never used the syntax [itex](dt)^2[/itex] so if I am in error, my apologies.
HallsofIvy said:I strongly suspect that you have misread
[tex]\int_0^x\int_0^x f(t)d\tau dt[/tex]
jackmell said:With all due respect Hall, I have not. It's from "A First Course in Integral Equations" by Abdul-Majid Wazwaz. It's on p. 20 and includes also, the expression:
[tex]\int_0^x \int_0^x \int_0^x f(t)dt dt dt=1/2 \int_0^x (x-t)^2 f(t)dt[/tex]
jackmell said:With all due respect Hall, I have not. It's from "A First Course in Integral Equations" by Abdul-Majid Wazwaz. It's on p. 20 and includes also, the expression:
[tex]\int_0^x \int_0^x \int_0^x f(t)dt dt dt=1/2 \int_0^x (x-t)^2 f(t)dt[/tex]
pwsnafu said:That's the Cauchy formula for repeated integration. And you are not allowed to write it as dtdtdt. You must keep the variables separate.
##\int_{a}^x \int_a^{t_1}\ldots \int_a^{t_{n-1}}f(t_n) \, dt_n \, dt_{n-1} \ldots dt_1 = \frac{1}{(n-1)!} \int_a^x (x-t)f(t) \, dt.##
AakashPandita said:I was trying to find the moment of inertia of a disk rotating at the axis through its center and perpendicular to it.
i got an integral for some function with (dx)^2 in it.
Why do you have "dm" in twice?AakashPandita said:[tex]I=\int_0^R dmr^2 dm=σπ[(r+dr)^2-r^2][/tex]
and then I put [tex]dm[/tex] in first.
Integrating with respect to (dx)^2 is useful in solving differential equations that involve higher-order derivatives. It allows us to find the general solution to the differential equation and identify any constants of integration.
Integrating with respect to (dx)^2 involves taking the integral of the square of the differential (dx)^2, while integrating with respect to dx involves taking the integral of the differential dx. The former is typically used when solving differential equations with higher-order derivatives, while the latter is used for simpler integrals.
The process for integrating with respect to (dx)^2 is similar to regular integration. First, we use the appropriate integration techniques (such as substitution or integration by parts) to simplify the integrand. Then, we integrate the resulting expression with respect to (dx)^2, treating it as a constant. Finally, we add any necessary constants of integration.
No, integrating with respect to (dx)^2 is typically only used for differential equations that involve higher-order derivatives. For simpler differential equations, integrating with respect to dx is usually sufficient.
Integrating with respect to (dx)^2 has various applications in physics and engineering, particularly in the study of vibrations and oscillations. It is also commonly used in the field of fluid dynamics to solve differential equations that model the motion of fluids.