Problem 2 - Thomas Calculus

Question

In Exercises \(1 - 8 :\) a. Set up an integral for the area of the surface generated by revolving the given curve about the indicated axis. b. Graph the curve to see what it looks like. If you can, graph the surface too. c. Use your utility's integral evaluator to find the surface's area numerically. $$y = x ^ { 2 } , \quad 0 \leq x \leq 2 ; \quad x -axis$$

Step-by-Step Solution

Answer

The surface area is approximately 47.409 units squared.

Step 1: Determine the Surface Area Formula

The surface area of a curve revolved around the x-axis is given by the formula \[S = \int_{a}^{b} 2\pi y \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx\]where \(y = x^2\), \(a = 0\), and \(b = 2\).

Step 2: Find the Derivative

Start by finding the derivative of the function with respect to \(x\). If \(y = x^2\), then \(\frac{dy}{dx} = 2x\).

Step 3: Substitute into the Formula

Substitute \(y = x^2\) and \(\frac{dy}{dx} = 2x\) into the surface area formula:\[S = \int_{0}^{2} 2\pi (x^2) \sqrt{1 + (2x)^2} \, dx = \int_{0}^{2} 2\pi x^2 \sqrt{1 + 4x^2} \, dx\]

Step 4: Graph the Curve and Surface

Graph the function \(y = x^2\) from \(x = 0\) to \(x = 2\). When this curve is revolved around the x-axis, it forms a paraboloid. Visualizing the curve and the surface can help understand the geometry involved.

Step 5: Evaluate the Integral Numerically

Use a numerical integration tool or calculator to evaluate the integral:\[\int_{0}^{2} 2\pi x^2 \sqrt{1 + 4x^2} \, dx\]You can use technology like a graphing calculator or software to find this value.

Key Concepts

Integral Calculus

Integral calculus is a branch of mathematics that deals with the concept of integration, a way to aggregate quantities that change continuously. One of the main objectives is to find the total or accumulated value, such as area, volume, or length. In this specific problem, we're using integral calculus to find the area of a surface of revolution. This is done by integrating along a curve that is rotated around an axis, creating a three-dimensional shape. The formula used here is for the surface area generated when a curve is revolved about the x-axis. This formula is: \[ S = \int_{a}^{b} 2\pi y \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \] Breaking it down, the integral sums up the infinite number of infinitesimally small bands that make up the surface area.

• \(y\) is the function value at a point, representing how far away from the x-axis the band is.
• \(\frac{dy}{dx}\) is the derivative, calculating how steep the function is at each point.
• The limits \(a\) and \(b\) indicate the interval over which we're calculating, in the problem given from 0 to 2.

Understanding this concept allows us to compute various physical properties such as volumes and surface areas in complex geometrical forms.

Numerical Integration

Numerical integration is a method used to approximate the value of integrals, especially when they are difficult or impossible to solve analytically. In practical scenarios, especially with complex integrals, exact solutions are hard to find. That's where numerical integration methods like Trapezoidal Rule, Simpson’s Rule, or using computational tools come into play.In this problem, the integral \[ \int_{0}^{2} 2\pi x^2 \sqrt{1 + 4x^2} \, dx \] doesn't have a simple antiderivative. This makes solving it analytically challenging, so we instead use numerical methods to estimate its value.

• Numerical integration breaks the area under the curve into simple shapes (like rectangles or trapezoids) to estimate the total.
• More refined methods use parabolic arcs for better accuracy, like Simpson's Rule.
• Modern computational tools can quickly compute these integrals, providing quick and accurate estimates for complex problems.

These methods are crucial in engineering, physics, and other applied sciences where practical computation is more important than symbolic mathematics.

Curve Sketching

Curve sketching is an essential skill in calculus that helps visualize the behavior of functions. By understanding the shape and direction of a curve, we can infer important properties about a function such as intercepts, turning points, and asymptotes. For the equation \(y = x^2\), a basic parabola, sketching the curve provides insight into its properties and how it behaves when revolved around an axis.When you're tasked to sketch the curve of \(y = x^2\) from 0 to 2:

• Plot key points, such as where \(x = 0\) gives \(y = 0\) and \(x = 2\) gives \(y = 4\).
• Notice symmetry. Parabolas like \(y = x^2\) are symmetrical along the y-axis.
• Understand the turn at \(x = 1\), which is halfway through the required plotting area.

When revolved about the x-axis, this curve creates a 3D shape called a paraboloid. Sketching helps not only in visualizing these shapes but also in accurately setting up mathematical integrals for areas and volumes.

Derivative of a Function

The derivative of a function represents the rate of change or slope of the function at any point. It provides vital information such as where the function is increasing or decreasing, and helps in the analysis of its curvature. In the context of finding the surface area of a revolution, calculating the derivative helps refine the integral setup by accounting for how the curve itself bends or inclines.For the given function \(y = x^2\), the process to find its derivative involves basic differentiation:

• The derivative \(\frac{dy}{dx}\) of \(y = x^2\) is \(2x\).
• This tells us that at any point \(x\) on the curve, the rate of change of \(y\) is directly proportional to the point's distance from the y-axis.
• In surface area calculations, this derivative term \(\left( \frac{dy}{dx} \right)^2\) becomes part of the integral to account for the incline of the curve.

Understanding derivatives is key in calculus as it enables us to work with dynamic systems and predict behaviors based on instantaneous rates of change.