Why is calculating the area under a curve required or rather what usage it would provide

The point is that in science and in real life, you come up against situations where you need to totalize or to aggregate or to accumulate some kind of growing quantity. Say you know how fast the snow is falling at any particular time, and want to know the accumulation after five hours. Say you have pollutants flowing into a lake at varying rates during the day or week, and want to know how much crud is in the lake after a few weeks. Both these examples are quantities that are accumulated from a varying contribution, and both are measured by an integral. And there’s no area in sight.

Based off of your question and the subsequent comments you have made I think I understand what you are looking for. I will give an example of what an integral can tell us, in addition to providing the area under a curve. Keep in mind, this is only one of very many examples.

We know that near the Earth's surface, an object in free fall accelerates at approximately $9.8 {m \over s^2}$. We can plot this acceleration as a function of time, and the resulting graph will look like a horizontal line. If I take the integral of this function from one point in time to another (say from $0$ to $10$ seconds), the result that I obtain is the change in velocity between those two points in time: $$ a(t) = 9.8$$ $$ \int_0^{10} 9.8dt=98 $$ What this result tells me is that if I release an object in the air and let it fall vertically for $10$ seconds it will undergo a change in velocity such that $\Delta v = 98 {m \over s}$. It also tells me the area from $t=0$ to $10$ under that horizontal line that is my acceleration graph, is $98$. As you can see, when I calculated this integral, I wasn't interested in finding the area under the acceleration graph. I was interested in finding the change in velocity. It just so happens to be that the two are the same thing; the area represents the change in velocity. This is one example of how the integral can tell you more than just the area under the graph.

The integral is used for far more than just computing areas of curves. However, if you do not have the intuition that comes from practicing with the idea in this form, you will have trouble applying integration in the future.

• Integrate jerk over time, get acceleration.
• Integrate acceleration over time, get velocity.
• Integrate velocity over time, get position.
• Integrate outward directed electrical flux over a surface, get enclosed charge.
• Integrate distance from an axis times density over space, get moment of inertia around that axis.
• Integrate blackbody radiation over frequency, get emitted power per square meter per steradian.
• Integrate magnetic field over a loop, get the current flowing through the loop.
• Integrate all possible future histories of a particle over the action, get its transition probability matrix (S-matrix).
• Integrate the wavefunction of a particle times the conjugate of a measurement over the phase space, get the probability of measuring that particle in that state.

... and those are just uses in Physics. There are many more in Mathematics.