Problem 2 - Thomas Calculus

Question

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

Step-by-Step Solution

Answer

The surface area is found through the integral \( \int_{0}^{2} 2\pi x^2 \sqrt{1 + 4x^2} \, dx \).

Step 1: Set up the Surface Area Integral Formula

To find the area of a surface generated by revolving a curve around an axis, we use the formula: \( A = \int_a^b 2\pi y \, ds \), where \( ds \) is the differential arc length and \( y \) is the function rotated around the axis.

Step 2: Calculate the Differential Arc Length

The differential arc length \( ds \) is given by the formula \( ds = \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \). For \( y = x^2 \), calculate \( \frac{dy}{dx} = 2x \). Then, \( ds = \sqrt{1 + (2x)^2} \, dx = \sqrt{1 + 4x^2} \, dx \).

Step 3: Plug into the Surface Area Formula

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

Step 4: Graph the Curve and Surface

Plot the curve \( y = x^2 \) from \( x = 0 \) to \( x = 2 \) on the coordinate plane. The curve is a parabola opening upwards. Visualize the surface by imagining this parabola rotated around the x-axis, forming a surface like a bell.

Step 5: Evaluate the Integral Numerically

Use a calculator or computational tool to evaluate the integral \( \int_{0}^{2} 2\pi x^2 \sqrt{1 + 4x^2} \, dx \). This will give the numerical value of the surface's area.

Key Concepts

Integral Calculus

Integral calculus is a branch of mathematics focused on the concept of integration, which is essentially the process of finding the area under a curve or the accumulation of quantities. In the context of surface areas of revolution, integration helps us determine the total area of a surface formed when a curve is revolved around an axis.
The general formula for the surface area is given by the integral \[A = \int_a^b 2\pi y \, ds\] Here,

• \(A\) is the surface area.
• \(y\) is the function describing the curve.
• \(ds\) is the differential arc length.
• \([a, b]\) is the interval over which the function is integrated.

When a curve such as \(y = x^2\) is rotated around the x-axis, this integral calculates the surface area of the resulting three-dimensional shape.
Integral calculus provides tools to analytically or numerically solve these integrals, leading to accurate and practical results for complex shapes and curves.

Differential Arc Length

Understanding differential arc length is crucial for calculating the surface area of a revolution. The differential arc length \(ds\) is an expression that represents the infinitesimally small segment of the curve. For a function \(y = f(x)\), the differential arc length is calculated as:
\[ds = \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx\] For the curve \(y = x^2\), the derivative \(\frac{dy}{dx} = 2x\) is first calculated. This derivative is then used in the arc length formula:
\[ds = \sqrt{1 + (2x)^2} \, dx = \sqrt{1 + 4x^2} \, dx\] By calculating \(ds\), we account for the curve's slope's effect on its length, ensuring an accurate surface area calculation when the curve is revolved. This understanding helps bridge geometric intuition with calculus, accurately representing a curve's complicated behavior.

Numerical Integration

Numerical integration is an essential technique in calculus that allows us to find approximate values for integrals when an antiderivative cannot be easily computed or expressed in simple terms. For complex functions like the integral for the surface area of revolution\[\int_{0}^{2} 2\pi x^2 \sqrt{1 + 4x^2} \, dx\]numerical methods provide valuable solutions. These methods might include:

• Trapezoidal rule
• Simpson's rule
• Monte Carlo integration

These methods involve breaking up the area under a curve into shapes (e.g., trapezoids or parabolic segments) whose exact areas can be calculated and summed to compute an approximate result.
Using computational tools or calculators equipped with numerical integration capabilities allows students to tackle integrals that are otherwise complex or impossible to solve analytically, enhancing problem-solving skills in applied mathematics.