Question

Estimate the area of the surface generated by revolving the curve \(y=2 x^{2}, 0 \leq x \leq 3\) about the \(x\) -axis. Use Simpson's rule with \(n=6\).

Short answer

The approximate area of the surface using Simpson's Rule with \( n=6 \) is \( 4\pi \) times the calculated integral value.

Step-by-step solution

Understanding the problem

We need to find the area of the surface generated by revolving the curve \( y = 2x^2 \) around the \( x \)-axis from \( x = 0 \) to \( x = 3 \). We will use the formula for surface area of revolution and employ Simpson's rule to approximate this surface area.

Surface of Revolution Formula

The formula for the surface area of revolution around the \( x \)-axis is: \[ S = 2\pi \int_a^b y \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \]Given \( y = 2x^2 \), we first need its derivative \( \frac{dy}{dx} = 4x \). Substitute \( y \) and \( \frac{dy}{dx} \) into the formula.

Substitute into the Integration Formula

Substitute into the formula:\[ S = 2\pi \int_0^3 2x^2 \sqrt{1 + (4x)^2} \, dx = 4\pi \int_0^3 x^2 \sqrt{1 + 16x^2} \, dx \] We need to approximate this integral using Simpson's Rule.

Setup for Simpson's Rule

To apply Simpson's Rule with \( n=6 \), calculate the step size \( h \) as follows: \[ h = \frac{b-a}{n} = \frac{3-0}{6} = 0.5 \] Then, identify the x-values: \( x_0 = 0, x_1 = 0.5, x_2 = 1.0, x_3 = 1.5, x_4 = 2.0, x_5 = 2.5, x_6 = 3.0 \).

Calculate Function Values

Calculate the function values at these points:\[ f(x_i) = x_i^2 \sqrt{1 + 16x_i^2} \]For instance:- \( f(x_0) = 0^2 \sqrt{1 + 16 \times 0^2} = 0 \)- \( f(x_1) = 0.5^2 \sqrt{1 + 16 \times 0.5^2} = 0.25 \sqrt{1 + 4} = 0.559 \ \)... (continue for x_2, ..., x_6)

Apply Simpson's Rule

Apply Simpson's Rule to approximate the integral:\[ \int_0^3 x^2 \sqrt{1 + 16x^2} \, dx \approx \frac{h}{3} [f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + 2f(x_4) + 4f(x_5) + f(x_6)] \]Calculate using the derived function values and \( h = 0.5 \).

Calculate the Surface Area

After calculating the Simpson's Rule approximation, multiply by \( 4\pi \) to get the surface area.For example, if the integral approximation using Simpson's Rule is \( I \), then surface area \( S = 4\pi I \). Substitute the result to get the final surface area's approximate value.

Key concepts

Simpson's Rule

Simpson's Rule is an efficient numerical integration technique that allows us to approximate the integral of a function over a closed interval. It's especially useful when the function does not have a straightforward antiderivative. The basic idea is to approximate the curve of the function by combining parabolas which fit over short segments of the curve.
To use Simpson's Rule effectively:

• Divide the interval of integration into an even number of subintervals, each of equal length.
• Identify the function values at each division point, including endpoints and midpoints within these subintervals.
• Apply the Simpson's Rule formula, which essentially weights these function values differently (1-4-2-4-...1 pattern), and aggregate these for the integral approximation.

In our problem, we divided from 0 to 3 into 6 equal subintervals. This means we calculate the function values at specific points, then use Simpson's Rule to approximate the integral, a critical step for eventual surface area calculation.

Surface Area Calculation

Calculating the surface area when a curve is revolved around an axis involves generating a surface of revolution. When set against the x-axis, this makes a 3D surface which can be measured using integral calculus.
The formula for the surface area of a revolution about the x-axis is:\[ S = 2\pi \int_a^b y \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \]This formula involves:

• The function to be revolved, \(y = 2x^2\) in our case.
• Derivative of the function, \(\frac{dy}{dx} = 4x\), indicating the curve's slope at any point.

These calculations involve substituting both the function and its derivative into the formula, creating an integral that represents the surface area. Our task then becomes approximating this integral which is where Simpson's Rule comes into play.

Integration Techniques

Integration techniques are essential tools in calculus for calculating areas, volumes, and other important quantities. In situations where antiderivatives are difficult to find manually, numerical methods like Simpson's Rule become valuable.
In our exercise, we are tasked with an integral that involves a square root function, \( x^2 \sqrt{1 + 16x^2} \). Exact integration here might not be straightforward. As such, numerical techniques simplify this process:

• Break down the integral range into small subintervals, facilitating easier approximation.
• Apply an integration method, such as Simpson's Rule, that averages the function's values over partitions, refining accuracy.
• Focus on a reliable numerical approximation when direct analytical methods are infeasible or overly complex.

Thus, while integration traditionally focuses on finding antiderivatives, numerical methods broaden its scope, making challenging surfaces solvable and the learning process more intuitive.