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Electric flux is a measure of the electric field passing through a given area. It's graphically represented by the number of electric field lines penetrating a surface. For our balloon scenario, imagine an invisible net catching the lines of the electric field generated by the charge inside. The total electric flux through our balloon's surface is expressed mathematically as the integral of the normal component of the electric field, denoted as , multiplied by the differential area element, dA, over the entire surface. In simple terms, you are summing up the electric field's strength at every tiny piece of the balloon's surface area. The outcome of 10 Nm²/C in our exercise indicates the total electric flux and directly relates to the charge inside under Gauss’s Law.

The enclosed electric charge refers to the total amount of charge residing within a closed surface. It is the source of electric fields that we detect outside the surface. In the context of Gauss's Law, it's the charge that creates the electric flux we have just discussed. To figure out the charge inside our hypothetical balloon, you must know the total electric flux passing through its surface. It's like knowing the amount of water flowing out from a sponge, and consequently, estimating how much water the sponge holds. In our mathematical journey, the step-by-step solution uses this principle to determine the balloon's hidden treasure – the amount of charge inside it.

Vacuum permittivity, denoted as _0, is a fundamental physical constant. It represents how much electric field is 'permitted' in the vacuum. Think of it as the measure of the vacuum's willingness to allow electric field lines to pass through itself. In terms of its role in Gauss's Law, it's a scaling factor that relates the electric flux through a surface to the enclosed charge. In essence, it ensures that the units balance out correctly, and we can calculate the enclosed charge based on the electric flux we observed. The value 8.85 × 10^-12 C²/Nm² is the accepted standard in physics and is crucial for the last steps of the solution where we determine the enclosed charge.

Integral calculus plays an essential role in physics, and Gauss's Law is a prime example of its application. It allows us to sum infinitesimally small pieces of a quantity to determine a total. In our scenario, we integrate or sum up the tiny contributions of the electric field over the balloon’s surface to obtain the total electric flux. It's like calculating the total amount of paint needed for a wall by adding up the paint required for each tiny section. The integral symbol ∮ represents this 'summing up' process over a closed surface, and it is the key to finding the solution to our problem. Integral calculus, thus, turns a conceptually challenging problem into a solvable puzzle using mathematical tools.