A resistor is made from a material that

Ohm's law is represented by the equation $I=\frac{V}{R}$. How would the current change if the amount of resistance decreased and the voltage stayed the same?

The formula for resistivity is:
$\rho =\frac{RA}{L}$
where $\rho$ is resistivity, $R$ is resistance, $A$ is cross-sectional area, and $L$ is the length of the conductor.
We can see from the formula that $A$ and $L$ are involved, why then does resistivity not depend on dimensions?

What is the resistivity coefficient of graphene?

A submarine of mass m is travelling at max power and has engines that deliver a force F at max power to the submarine. The water exerts a resistance force proportional to the square of the submarine's speed v.
The submarine increases its speed from $v_1$ to $v_2$, show that the distance travelled in this period is $\frac{m}{2k}\ln \frac{F-k{v}_1^2}{F-k{v}_2^2}$ where k is a constant.
I've found $\frac{dv}{dt}=\frac{1}{m}(F-k{v}^2)$ using $\sum F=ma$ but I am struggling to progress further using integration or the fact that $\frac{dv}{dt}=v\frac{dv}{dx}=\frac{d}{dx}(\frac{1}{2}{v}^2)=\frac{d^{2}x}{d{t}^2}$ which is how I've been taught to solve resisted motion questions.

Say we got an alloy made of $90$% Au and 10%Cu. The atomic mass of gold is
$A_{Au}=197g/mole$
and of copper
$A_{Cu}=63.5g/mole$
The specific resistivity of gold is
$\rho =22.8n\mathrm{\Omega }m$
and the Nordheim coefficient is
$C=450n\mathrm{\Omega }m$
Then by the Nordheim rule we have that the resistivity of this alloy is given as
$\rho ={\rho }_{Au}+CX(1-X)$
where X=0.1, and represents the percentage of Copper in the alloy. Using the above I got that
$\rho =63.3n\mathrm{\Omega }m$
, but data shows that the actual resistivity is $\rho =108n\mathrm{\Omega }m$. What about X? Is it the percentage of atoms in the alloy, or is it the percentage of mass in the alloy?

$V_P-V_O={\int }_P^{O}E.dr$
The equation is in one dimension and P and O are points on the x axis.
The Electric field is oriented in the $+x$ direction ans has the formula $c\frac{1}{r}$ where c is a coefficient and r is the distance form the origin of the coordinate system.
The vector ${\overrightarrow{r}}_P-{\overrightarrow{r}}_O$ has the same direction as that of the electric field.
Now, Why do we put P below the integral and O above it? Can someone explain the equation for me?