# How to apply Leibniz's Rule to a Metal Pipe Temperature's Partial Derivative in this Example

Refer to this image showing the temperature of a metal pipe at the inlet and the outlet:

The temperature $$T(z,t)$$ is a function of the length $$z$$ and time $$t$$. Let the average temperature be
$$T_mathrm{avg}(t)=frac{1}{2}big(T(a,t)+T(b,t)big).$$
Integrating the partial derivative of $$T$$ with respect to $$t$$, $$frac{partial T}{partial t}$$ over the entire length of the pipe (from $$z=a$$ to $$z=b$$) and applying Leibniz’s rule, we should get
$$int_{z=a}^{z=b}frac{partial}{partial t}T(z,t)dz=frac{d}{dt}int_{z=a}^{z=b}T(z,t)dz-T(b,t)frac{db}{dt}+T(a,t)frac{da}{at}.$$
In this paper (eqs. (21) and (22)), the result is reported as follows:
$$int_{z=a}^{z=b}frac{partial}{partial t}T(z,t)dz=(b-a)frac{d}{dt}T_mathrm{avg}(t)+big(T(a,t)-T_mathrm{avg}(t)big)frac{da}{dt}+big(T_mathrm{avg}(t)-T(b,t)big)frac{db}{dt}.$$

Are the two equivalent? How to correctly solve this integral? Thanks!

Note: The integral in this question is only a portion of a larger integration problem in the energy balance, not reported for brevity. Equations (21) and (22) in the reference describe the complete energy balance.

This answer is based on the original answer here. Some supplemental details relevant to the problem are added.

Referring to the original problem, let $$T(z,t)$$ vary linearly with $$z$$ as shown in the following figure:

As proposed in the original solution, integrating integral of the partial derivative of $$T$$ from $$a$$ to $$b$$ and applying Leibniz's Rule: begin{align} int_{a}^{b}frac{partial}{partial t}T(z,t)dz&=int_{a}^{b}frac{partial}{partial t}big(T(z,t)-T_mathrm{avg}(t)+T_mathrm{avg}(t)big)dz\ &=int_{a}^{b}frac{partial}{partial t}T_mathrm{avg}(t)dz + underbrace{int_{a}^{b}frac{partial}{partial t}big(T(z,t)-T_mathrm{avg}(t)big)dz}_text{apply Leibniz Rule}\ &=(b-a)frac{d}{dt}T_mathrm{avg}(t)\ &quad+underbrace{frac{d}{dt}int_{a}^{b}big(T(z,t)-T_mathrm{avg}(t)big)dz+big(T(a,t)-T_mathrm{avg}(t)big)frac{da}{dt}+big(T_mathrm{avg}(t)-T(b,t)big)frac{db}{dt}.}_text{Leibniz's Rule applied} end{align}

The area under the curve in the figure (shaded green) can be found using the integral $$int_{a}^{b}T(z,t)dz$$. Approximating the curve with a straight line, we can write

begin{align} int_{a}^{b}T(z,t)dz&approxunderbrace{frac{1}{2}(b-a)big(T(b,t)-T(a,t)big)}_text{area of top triangle}+underbrace{T(a,t)(b-a)}_text{area of bottom rectangle}\ &=frac{1}{2}big(T(a,t)+T(b,t)big)(b-a)=(b-a)T_mathrm{avg}(t). end{align}

Thus, the second term in the third equality of the first equation vanishes, and the desired result is obtained: $$int_{a}^{b}frac{partial}{partial t}T(z,t)dzapprox(b-a)frac{d}{dt}T_mathrm{avg}(t)+big(T(a,t)-T_mathrm{avg}(t)big)frac{da}{dt}+big(T_mathrm{avg}(t)-T(b,t)big)frac{db}{dt}.$$

## Related Questions

### Find function which minize variance of intensity on target plane

0  Asked on January 12, 2021

### Lagrangian of the Stretching Mode Vibration of the Acetylene Molecule

1  Asked on January 12, 2021

### Are orbiting masses in a uniform disc affected by masses outside its orbit?

3  Asked on January 12, 2021

### The nature of the electromagnetic induction

2  Asked on January 12, 2021 by user157308

### Electron in free space and Schrodinger’s equation

4  Asked on January 12, 2021 by anwesa-roy

### Do smaller organisms (eyes) see smaller objects in greater clarity?

1  Asked on January 12, 2021

### Is there a way to split a black hole?

4  Asked on January 12, 2021 by lurscher

### Supersymmetry non-breaking $iff$ no “Goldstone fermion”?

0  Asked on January 12, 2021

### Calculating the redshift drift

1  Asked on January 12, 2021

### Conductor with hollow cavity under external electric field

1  Asked on January 12, 2021 by user3001408

### How to calculate impulse at a particular time instant?

1  Asked on January 12, 2021 by infinitecool23

### Dealing with degeneracy in Paschen-Back Effect

1  Asked on January 12, 2021 by user148792

### Using Helmholtz Free Energy to Calculate Liquid Density

1  Asked on January 12, 2021 by sam-o

### Physical meaning of eigenvalues in the heat equation problem

1  Asked on January 12, 2021 by jakub-korsak

### Polar radius and position vector: two-dimensional kinematics for high school students

2  Asked on January 12, 2021

### What is electron precipitation?

1  Asked on January 12, 2021 by ceruwof

### Why is the magnetic field dot producted in the integral version of Ampere’s circuital law?

2  Asked on January 12, 2021

### Does increasing the air density change always has an effect of a partly submerged object in liquid system?

1  Asked on January 12, 2021 by user359455

### Goldstone bosons in soft limit

1  Asked on January 12, 2021