For the curve \(y=\sqrt{x},\) between \(x=0\) and \(x=2,\) find: The volume of the solid generated when the area is revolved about the \(x\) axis.

Integral calculus is one of the two main branches of calculus, the other being differential calculus. It focuses on the concept of the integral, which is a mathematical tool used to calculate areas, volumes, and other related quantities. In simpler terms, while differential calculus deals with rates of change (like velocity of a car), integral calculus deals with accumulation of quantities (like distance covered by a car).

Integral calculus is deeply connected with the concept of the antiderivative. An integral essentially 'undoes' what a derivative does. There are two main types of integrals:

• Indefinite Integral: Represents a family of functions and includes a constant of integration (C).
• Definite Integral: Represents a number, which usually indicates the area under a curve or the total accumulated quantity.

To understand integral calculus, one must grasp how to set up and evaluate these integrals. This includes knowing how to apply limits, work with infinite sums, and use properties like linearity. Integrals play an essential role in many scientific fields, such as physics, engineering, and economics.

A definite integral is a type of integral that calculates the net area under a curve within a specified range. More formally, for a continuous function f(x) on the interval \(a, b\), the definite integral is written as \[ \int_{a}^{b} f(x) \, dx \]

In this exercise, we used a definite integral to find the volume of a solid of revolution. To compute the definite integral, the following steps are typically followed:

• Set up the integral by determining the limits of integration (from \(a\) to \(b\)).
• Identify the function to integrate (in this case, \(f(x) = \sqrt{x} \)).
• Evaluate the integral using proper techniques (substitution, integration by parts, etc.)
• Compute the limits and subtract to get the net result.

For instance, the net volume of the solid formed by revolving the function \( y = \sqrt{x} \) around the x-axis, across \(x = 0\) to \(x = 2\), was obtained using the definite integral \[ \pi \int_{0}^{2} \left(\sqrt{x}\right)^2 \, dx \]. This integral quantified the collective volume within specified boundaries.

A solid of revolution is a three-dimensional shape generated by rotating a two-dimensional curve around an axis. This concept is widely used in both mathematics and physics to compute volumes of various objects. When revolving a curve around the x-axis or y-axis, the resulting solid’s volume can be calculated using techniques from integral calculus.

To find the volume of a solid of revolution about the x-axis, the formula used is \[V = \pi \int_{a}^{b} [f(x)]^2 \, dx\]. This formula stems from the idea of summing up infinitely many thin disk volumes, each with radius \(f(x)\) and thickness \(dx\).

This method applies to many different curves and situations. For instance, in our exercise, the curve \(y = \sqrt{x} \) was revolved around the x-axis to form a solid. By integrating its squared value \sqrt{x} \, over the interval from \(x = 0\) to \(x = 2\), we obtained the total volume as \[V = 2 \pi \]

Understanding solids of revolution can be vital in numerous practical applications, ranging from engineering designs to volume-based economic models. It allows us to transform simple curves into complex three-dimensional objects.