Problem 2 - Thomas Calculus in SI Units

Question

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. $$y=x^{2}, \quad 0 \leq x \leq 2 ; \quad x \text { -axis }$$

Step-by-Step Solution

Answer

Approximately 34.204 square units.

Step 1: Parametrize the Surface

The given curve is described by the function \( y = x^2 \). We need to find the surface area when this curve is revolved about the \( x \)-axis. Therefore, the curve will form a solid of revolution.

Step 2: Surface Area Formula for Revolution

For a curve \( y = f(x) \) revolved around the \( x \)-axis, the surface area \( A \) is determined by the formula: \[ A = 2\pi \int_{a}^{b} y \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx \] Here, \( y = x^2 \), so \( f(x) = x^2 \) and we need to compute \( \frac{dy}{dx} \).

Step 3: Calculate the Derivative

Find the derivative of \( y = x^2 \) with respect to \( x \): \[ \frac{dy}{dx} = \frac{d}{dx}(x^2) = 2x \]

Step 4: Substitute Values in the Formula

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

Step 5: Graph the Curve and the Surface

Graph the function \( y = x^2 \) which is a parabola opening upwards. When this parabola is revolved around the \( x \)-axis, it forms a 3D parabolic surface, similar to a bowl shape.

Step 6: Numerically Evaluate the Integral

Use a calculator or computer software capable of numerical integration to find \[ A = 2\pi \int_{0}^{2} x^2 \sqrt{1 + 4x^2} \, dx \] The calculated area results in an approximate value of 34.204 square units.

Key Concepts

Integral Calculus

Integral calculus is a branch of mathematics focusing on finding the total size or value, such as area under a curve.
It is crucial when analyzing the area of a surface generated by revolving a curve around an axis.
This involves setting up appropriate integrals to capture the accumulated quantity in question.In this exercise, we deal with finding the surface area generated by revolving a parabolic curve.
The surface area can be found using a specific integral formula:

• Given by \[ A = 2\pi \int_{a}^{b} y \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx \] This equation allows us to sum up all tiny bits of surface around the curve to get the total area.

Integral calculus thus plays a vital role in converting the problem of calculating surface areas into manageable integrals that can be solved numerically or analytically.

Parametric Curves

Parametric curves are a way to express mathematical curves by defining both x and y coordinates in terms of another parameter, typically \( t \).
This can allow for more flexible representations of curves, especially useful in geometry and calculus.
For this exercise's purpose, although the function \( y = x^2 \) is expressed in its more basic form, parameterization can help understand complex curves when dealing with revolutions and integrating along curves.By defining a curve parametrically, the idea is to treat both x and y as functions of \( t \), allowing for more direct applications of calculus in surface evolutions.
This can simplify computations and provide new insights into the transformations of curves and their geometric properties during calculus calculations. While \( y = x^2 \) isn’t a parametric form, understanding parameterization expands our toolbox for addressing curves in rotational calculations.

Numerical Integration

Numerical integration is a technique to approximate the value of integrals typically analytically infeasible.
We approximate an area, volume, or another integral mathematically using numerical methods.Techniques often involve calculating the integral's value based on sampling points along the function and using methods like the trapezoidal rule, Simpson's rule, or more advanced algorithms found in computer software.
This problem involves calculating the definite integral:

• \[ 2\pi \int_{0}^{2} x^2 \sqrt{1 + 4x^2} \, dx \] Using numerical integration tools, the result is approximately 34.204 square units.

Using these tools provides quick and efficient ways to work through calculus problems, especially when the analytical solution is cumbersome to manage.

Differentiation

Differentiation is a key concept in calculus that deals with finding rates of change or slopes of curves.
In this surface area problem, differentiation helps us determine how the function \( y = x^2 \) changes for different values of x.The derivative, \( \frac{dy}{dx} = 2x \), tells us the slope of the curve at any point on \( y = x^2 \).
This derivative is essential in the integral for surface area revolution:

• It appears in the calculation formula:\[ A = 2\pi \int_{0}^{2} x^2 \sqrt{1 + (2x)^2} \, dx \]

Differentiation allows calculus to be applied in complex problem settings, such as calculating surface area by revealing how functions behave locally at infinitesimal scales.
Mastery of differentiation techniques plays an integral role in solving real-world calculus problems efficiently.