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Integral calculus is one of the two principal branches of calculus, with the other being differential calculus. It deals with the accumulation of quantities, such as areas under curves, total distance traveled, or accumulated change. When we seek to find the area under a curve between two points on a graph, we are tackling a fundamental problem in integral calculus. This process involves determining a function's integral, which gives the accumulation of the function's rate of change.

The importance of integral calculus cannot be overstressed as it is pivotal in various fields of science and engineering. For instance, it helps physicists calculate the center of mass, engineers to determine work done by a force, and economists to model accumulated growth over time. By understanding the integral as an 'accumulator,' students can grasp the practical significance of integral calculus beyond the mathematical procedures.

Approximating the area under a curve is crucial when an exact solution is difficult to obtain or when a quick estimate is needed. In practice, finding the exact area under a curve on a graph involves calculating the definite integral, but when this isn't feasible, approximation methods come into play.

Two such methods are the use of upper and lower sums. By dividing the domain of the function into equal sub-intervals, we can create rectangles that either overestimate or underestimate the actual area. The sum of the areas of these rectangles gives us an approximation of the integral. If the function is monotonically increasing, the left endpoint of each sub-interval will give the lower sum, and the right endpoint will give the upper sum. Conversely, if the function is monotonically decreasing, the roles of the endpoints will swap. The finer the subdivision (i.e., the more rectangles we use), the closer the approximation will be to the exact area.

Definite integrals are a way to denote the exact area under a curve from one point to another on a graph. Written in the form of \( \int_{a}^{b} f(x) \,dx\), where \(a\) and \(b\) represents the interval's endpoints, and \(f(x)\) is the function being integrated, the definite integral quantifies the net area between the function and the x-axis.

The fundamental theorem of calculus links definite integrals and antiderivatives, showing that evaluating the antiderivative at the endpoint of an interval \(b\) and subtracting its evaluation at \(a\) gives the definite integral's value.

Referring to our example, the calculation \(0.6 < \int_{0}^{1} \frac{dx}{1+x^{2}} < 1\) demonstrates the use of a definite integral. It is given bounds, which, after evaluation, suggest that the area under the curve \(y=\frac{1}{1+x^{2}}\) from \(x=0\) to \(x=1\) is greater than 0.6 but less than 1. This precise quantification is the essence of the concept of definite integrals in integral calculus.