91Ó°ÊÓ

A definite integral can be thought of as a way to calculate the area under a curve between two points. In mathematics, it is usually represented in the form \( \int_a^b f(x) \, dx \), where \(a\) and \(b\) are the limits of integration, and \(f(x)\) denotes the function being integrated. The result of a definite integral is a number, which represents the total area under the curve from \(x = a\) to \(x = b\).
In the context of surfaces of revolution, the definite integral helps us find the surface area of a 3D object created by rotating a curve around the \(x\)-axis. Here, the bounds \(a = 0\) and \(b = 3\) define the interval over which the curve \(y = 3x\) is revolved, allowing us to calculate the surface generated.

When calculating this area, you integrate the `surface area formula`, which involves a combination of the function and its derivative under a square root sign. Applying definite integration, as in the solution, is essential for achieving the final numerical result.

Differentiation is a fundamental concept in calculus, concerning the rate at which a function changes. In the exercise, given the function \(y = 3x\), differentiation requires us to find its derivative, \(\frac{dy}{dx}\). For linear functions like \(y = 3x\), we use the power rule: for \(f(x) = cx^n\), the derivative is \(\frac{d}{dx}[cx^n] = ncx^{n-1}\).
By applying this rule, \(y = 3x\) simplifies to \(\frac{dy}{dx} = 3\), since the derivative of \(x\) with respect to \(x\) is 1, and multiplying by 3 gives 3.

In the problem of finding a surface area of revolution, the derivative \(\frac{dy}{dx}\) is crucial. It's part of the formula \(2\pi \int_a^b y \sqrt{1 + (\frac{dy}{dx})^2} \, dx\). Here, \(\frac{dy}{dx} = 3\) helps in calculating the required surface area, as it affects the shape of the curve when rotated and thus the surface of the resulting solid.

Integration is the inverse process of differentiation and is about finding a function whose derivative corresponds to the given function, or finding the total accumulation of quantities.
In our exercise, once the parameters are set using differentiation, integration comes into play to calculate the area of the surface of revolution. Specifically, you perform the integral \(2\pi\sqrt{10} \int_0^3 3x \, dx\).
To simplify your work, utilize the constant factors outside the integral sign. In this case, \(3\) and \(\sqrt{10}\) come out, making the integral easier to handle. By integrating \(3x\), you find \(\frac{3}{2}x^2\), and thus, the antiderivative becomes easy to evaluate between 0 and 3.

This step transforms your function into a calculated value, representing the surface area obtained by rotating \(y = 3x\) from \(x = 0\) to \(x = 3\) around the \(x\)-axis.

The surface area formula for a surface of revolution is an essential tool in this exercise's solution. It is given by the integral \(2\pi \int_a^b y \sqrt{1 + (\frac{dy}{dx})^2} \, dx\). Used to calculate the surface area of a rotated curve, this formula accounts for the shape and length of the curve.
Breaking it down:

• \(y\) refers to the function that describes the curve, in this case \(3x\).
• \(\frac{dy}{dx}\) is the derivative of \(y\), simplifying to \(3\).
• The term \(\sqrt{1 + (\frac{dy}{dx})^2}\) accounts for the change in the curve's slope, ensuring all aspects of its shape are considered.
  
  By inserting the curve's function \(y = 3x\) and calculating the derivative \(\frac{dy}{dx}\), you use the formula to numerically estimate the total outer surface formed by revolving the curve around the \(x\)-axis. The result, \(27\pi \sqrt{10}\), is the exact surface area for the situation described.