Introduction

Before diving into calculus and memorizing formulas, it is important to take a step back and understand what an integral actually means.

What is an Integral?

An integral, graphically, represents the area under a curve. You might be wondering why it’s useful to know the area under the curve. The reason is that the area under the curve represents the accumulation of the quantity described by the function over an interval.

For example, if we have a simple graph that charts book sales per hour, the integral of that graph, or area under the curve, gives the total number of books sold over a period of time. By looking at the rate of sales, we can find the total accumulation of sales.

Applications in Physics

This concept has many applications, one of the most common being in physics. Velocity is the rate of change of distance. The area under the velocity-time curve represents the total distance traveled.

Suppose \( v(t) \) represents velocity as a function of time. Then the distance \( s \) traveled from time \( t = a \) to \( t = b \) is:

\[\begin{align*} s = \int_{a}^{b} v(t) \, dt \end{align*}\]

By calculating this integral, we sum up all the small distances covered over each infinitesimal time interval between \( a \) and \( b \).

Conclusion

Understanding integrals as the area under a curve allows us to solve real-world problems involving accumulation, such as calculating total sales, distance traveled, or any other quantity that accumulates over time.