Change of magnetic flux on size of magnetic field and area Consider a solenoid with 50 turns. Each turn gives has an area $A$. When the power is switched on, each turn produces a magnetic field of 1 Tesla. What is the total change in magnetic flux when the power is switched on?
I know that $\Phi=\iint\vec{B}.\vec{A}$.When the power is switched on the total magnetic field is 50 Tesla. But how about the area? Is it necessary to multiply the area by 50 times? Or the definition of flux is only considering a point?
 A: The flux will be 50 times the (area of each turn)$*$50 Tesla-m$^2$.
The flux is defined considering an area i.e we consider an area of a closed loop and then calculate the magnetic flux through it;so there is no question of a point.
In your question,the flux through each of the turns is 50 times the area of each turn,and then to find the total flux we multiply by 50,since there are 50 of those turns.
Professor W.Lewin has explained this beautifully.Imagine a metal rung of 50 turns.You dip it inside a soap solution and take it out.Well,now you can imagine what will happen(imagining is quite hard,honestly:P).You are gonna have 50 layers of soap sol.  with each turn and you consider each soap layer to be your area. Now if it is kept in a magnetic field ,the magnetic field will have to pass through 50 soapy layers,which gives you explanation why you multiply by 50.
A: Consider a circular circuit with a uniform magnetic field of magnitude B passing perpendicularly through the surface defined by the circuit and has area A. For the sake of the analogy I am about to make, consider the surface to be the simplest: a disk. The magnetic flux is simply B*A.
Now, lets say that the solenoid is a circuit by itself (just connect a battery to its two ends). Now say have the same magnetic field passing through the area that is defined by the solenoid. But wait a minute, in the first example we had a circle which had an easily defined area through which the magnetic field was normal to but in this case the area is much more complicated to visualize. I give a picture of it:

Now, your imagination is needed. To find the magnetic flux, we find the area that the magnetic field is perpendicular to. Thus, to find the magnetic flux in this case, we need to find the "component" of the area that is perpendicular to the magnetic field(if you know about surface orientation, you will understand what components of the area means, although it is an informal description I think).
We do this by projecting the surface defined by the solenoid on a plane that is perpendicular to the magnetic field:

The projection that we get is a disk with radius being that of the loops of the solenoid (the same as the disk defined by the circuit of the first example). But, for the magnetic flux we need to know how many times a disk is projected. Well, with some imagination, you can conclude by yourself that it is as many times as there are loops. So, in this case, it's 50.  
Keep in mind, though, that this projection of the surface is not mathematically necessary in order to obtain this result. I am just pointing it out so it becomes easier for you to imagine the situation and be motivated to just multiply B*A of the first example by 50 times.  
So, to conclude, it's just 50 B*A (as you correctly concluded in your question, it's 50 Tesla in this case).
(The image depicting the area defined by the solenoid is found on the cover of Zill's Introduction to Complex Analysis)
A: You asked: "What is the total change in magnetic flux when the power is switched on?" Technically that question can't be answered because "flux" is the dot product of the vectors of some vector field (in this case the magnetic field) with the normal vector of some surface, integrated over that surface. So you integrate over the surface the dot product of the vector field and the normal to get a flux. So the person answering must ask "Over what surface?", before they can answer. 
It is usually understood when discussing solenoids that the area in question is a cross section of the solenoid perpendicular to its longitudinal axis. It is also usually assumed that the displacement of each coil along the axis of the solenoid is negligible. In that case each coil adds an equal increase in the magnetic field through the single disk of the cross section. Integrating the field dotted with the normal vectors over that disk sums the effects of each coil and since there is only one disk you don't need to multiply again. Each coil contributes equally to the magnetic field through a single disk perpendicular to the axis of the solenoid with radius equal to the radius of the solenoid and the flux of the magnetic field through it once transients have died is what is being asked for. Answer: only one area. 
You can imagine many other areas and also calculate the fluxes through them also. You cannot really sum over all possible areas, or if you do the answer would be infinite, because you can imagine an infinite number of areas in any volume of space and the magnetic field exists at each point in 3d space. Generally the flux through an area gets less if you consider areas away from the coil, but again usually a single disk through the center of the coil is meant.
