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 grapher's or computer'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

Surface area: numerically evaluated using \( \int_{0}^{2} 2\pi (x^2) \sqrt{1 + 4x^2} \, dx \).

Step 1: Identify the formula for surface area of revolution

The surface area of a curve revolved around the x-axis is calculated using the formula \(S = \int_{a}^{b} 2\pi y \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx\). For the given curve \(y = x^2\), we first need to find \(\frac{dy}{dx}\).

Step 2: Compute the derivative

We have \( y = x^2 \). The derivative of \( y \) with respect to \( x \) is \( \frac{dy}{dx} = 2x \).

Step 3: Set up the integral

Substitute \( y = x^2 \) and \( \frac{dy}{dx} = 2x \) into the formula: \[ S = \int_{0}^{2} 2\pi (x^2) \sqrt{1 + (2x)^2} \, dx \]. Simplify the expression under the square root to obtain \( \sqrt{1 + 4x^2} \).

Step 4: Graph the curve y = x^2

Plot the curve \( y = x^2 \) for \( 0 \leq x \leq 2 \). It is a parabola opening upwards with the vertex at the origin (0,0).

Step 5: Graph the surface of revolution

The surface is generated by rotating the parabola about the x-axis, forming a 3D shape resembling a wide funnel opening along the x-axis.

Step 6: Evaluate the integral numerically

Using a graphing calculator or computational tool, evaluate the integral \[ S = \int_{0}^{2} 2\pi (x^2) \sqrt{1 + 4x^2} \, dx \] to find the numerical approximation for the surface area.

Key Concepts

Integral Calculus

Integral calculus is a major part of calculus that focuses on the concept of integrals. It plays a vital role in finding areas, volumes, and other quantities that are essential in mathematics and its applications. In this exercise, we explore the surface area of revolution, which is an application of integral calculus.

To find the surface area of a curve rotated around an axis, we use the formula for the surface area of revolution. For the problem statement, the revolved curve is \(y = x^2\) about the x-axis. Here, our job is to calculate the integral which represents the surface area. The process involves substituting the function and its derivative into the formula for the surface area of revolution.

Definite Integral

A definite integral is used to compute the exact value of the area under a curve between two specified limits. In our exercise, we aim to solve the definite integral to determine the surface area formed by revolving the curve \(y = x^2\) from \(x = 0\) to \(x = 2\) around the x-axis.

The function is substituted into the formula:

• The surface area is expressed as \( S = \int_{0}^{2} 2\pi (x^2) \sqrt{1 + (2x)^2} \, dx \).

This definite integral calculates the area covered as the curve sweeps around the axis, accounting for the whole interval from 0 to 2.

Numerical Integration

Numerical integration refers to methods used to approximate the value of an integral, especially when an analytical solution is difficult or impossible to find. In this exercise, after setting up the definite integral for the surface area, numerical integration becomes essential to approximate the integral \( S = \int_{0}^{2} 2\pi (x^2) \sqrt{1 + 4x^2} \, dx \).

As this problem might involve complex calculations due to the square root component, we can turn to tools like graphing calculators or computer software:

• These tools apply methods such as Simpson's Rule or Trapezoidal Rule.
• This enables us to get a numerical approximation for the desired surface area.

Derivative

The concept of a derivative is central to calculus, measuring how a function changes as its input changes. For this problem, we had to first find the derivative of \(y = x^2\).

The derivative \(\frac{dy}{dx} = 2x\) was calculated and used because it describes the slope of the curve at any given point. When solving surface area problems, the derivative is included in the formula to account for the rate of change along the axis of rotation.

• This means: it helps us express the steepness of the curve during the revolution process
• It also factors into the integral's setup by adjusting the curve's path length (or arc length).