Problem 57 - Calculus III

Question

The surface of a large cup is formed by revolving the graph of the function \(y=0.25 x^{1.6}\) from \(x=0\) to \(x=5\) about the \(y\) -axis (measured in centimeters).[1] Use technology to graph the surface.

Step-by-Step Solution

Answer

Use the formula for surface area of revolution around the y-axis and compute the integral numerically to graph the surface.

Step 1: Set up the surface of revolution

To find the surface of revolution, we need to use the formula for the surface area obtained by revolving a curve around the y-axis. The formula for this is \[ S = 2\pi \int_{a}^{b} x \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \; dx \] where \( y = 0.25x^{1.6} \).

Step 2: Calculate the derivative

First, we need to find the derivative \( \frac{dy}{dx} \) of the function \( y = 0.25x^{1.6} \). Using the power rule, the derivative is: \[ \frac{dy}{dx} = 0.25 \times 1.6 x^{0.6} = 0.4 x^{0.6} \]

Step 3: Substitute into the formula

Substitute \( y = 0.25x^{1.6} \) and \( \frac{dy}{dx} = 0.4 x^{0.6} \) into the surface area formula: \[ S = 2\pi \int_{0}^{5} x \sqrt{1 + (0.4x^{0.6})^2} \; dx \] Simplifying further gives us: \[ S = 2\pi \int_{0}^{5} x \sqrt{1 + 0.16x^{1.2}} \; dx \]

Step 4: Use technology for evaluation

Use technological software or a calculator with numerical integration capability to compute the integral: \[ S = 2\pi \int_{0}^{5} x \sqrt{1 + 0.16x^{1.2}} \; dx \] This gives a numerical approximation of the surface area of the cup when evaluated.

Key Concepts

Surface Area of Revolution

To understand the concept of surface area in the context of revolutions, imagine you have a curve, and by revolving this curve around an axis, you create a 3D object. The surface area is the total outer layer of this solid. In the given exercise, a function is revolved around the y-axis, generating the surface or the outside of a cup.
Utilizing the mathematical formula for surfaces generated by revolving a curve can help calculate this area. Specifically, when a curve is rotated around the y-axis, we use:

• \( S = 2\pi \int_{a}^{b} x \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \; dx \)

This formula requires knowledge about both integration and derivatives, which we will cover in later sections. Understanding surface of revolution helps us possibly model real-world objects like vases or cups, further bridging practical applications with mathematics.

Integration in Calculating Surface Area

Integration acts as a powerful tool in calculating areas under curves, among other applications. When we talk about surfaces of revolution, integration helps sum up an infinite number of infinitely thin rings (cross-sections of the revolved function) to find the total surface area.
The integral \( \int_{a}^{b} x \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \; dx \) symbolizes this summation process. Here, each tiny slice is rotated around the axis, forming a ring whose circumference contributes to the surface area.
By setting boundaries from \(x=0\) to \(x=5\), we define the limits of integration, encapsulating only the desirable section of the curve that contributes to the surface. Integration, hence, acts as a bridge, turning the abstract formula into tangible surface area results when calculated effectively.

The Role of Derivatives

In the realm of calculus, derivatives signify the rate of change of functions. Within our exercise, derivatives provide the slope of the function \(y=0.25x^{1.6}\), representing how steep or flat the curve is at any given point. Knowing this slope is crucial as it's part of the integral inside our surface area formula.
To find the derivative of the function, apply the power rule, which involves multiplying the power by the coefficient and then subtracting one from the exponent:

• Given \( y = 0.25x^{1.6} \), its derivative \( \frac{dy}{dx} = 0.4 x^{0.6} \).

Understanding derivative helps to precisely model the twist and turn of the curve, further impacting the calculus of surface through rotation.

Numerical Approximation of Integrals

Numerical approximation offers a practical solution when integrals are too complex to solve analytically. It involves estimating the value of the integral using technology or methods such as Simpson's Rule or the Trapezoidal Rule.
For our cup surface problem, evaluating the integral

• \( 2\pi \int_{0}^{5} x \sqrt{1 + 0.16x^{1.2}} \; dx \)

exactly is challenging, hence the reliance on numerical methods. Modern software or scientific calculators can provide close estimates by approximating the area under the curve through subdivisions and calculating smaller, manageable segments efficiently.
This approach gives an accurate enough approximation for practical purposes without delving into potentially complicated symbolic solutions, making complex integrals accessible even without advanced calculus expertise.