Problem 5 - 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. \(x^{1 / 2}+y^{1 / 2}=3\) from \((4,1)\) to \((1,4) ; \quad x\) -axis

Step-by-Step Solution

Answer

Evaluate the integral numerically to find the surface area.

Step 1: Simplify the Equation

Begin by simplifying the given equation \( x^{1/2} + y^{1/2} = 3 \). Solve for one of the variables; let's solve for \( y \). This gives us \( y^{1/2} = 3 - x^{1/2} \). Square both sides to find \( y \) in terms of \( x \): \( y = (3 - x^{1/2})^2 \).

Step 2: Determine the Derivative

To set up the integral for the surface area, we need the derivative \( \frac{dy}{dx} \). Using \( y = (3 - x^{1/2})^2 \), apply the chain rule. Let \( u = 3 - x^{1/2} \), so \( y = u^2 \). Then \( \frac{du}{dx} = -\frac{1}{2}x^{-1/2} \), and \( \frac{dy}{du} = 2u \). Thus, \( \frac{dy}{dx} = 2(3-x^{1/2})(-\frac{1}{2}x^{-1/2}) \), which simplifies to \( -\frac{(3-x^{1/2})}{x^{1/2}} \).

Step 3: Set Up the Integral for Surface Area

The formula for the surface area of a curve revolving around the x-axis is \( 2\pi \int_{a}^{b} y \sqrt{1+\left(\frac{dy}{dx}\right)^2} \, dx \). Substitute \( y = (3-x^{1/2})^2 \) and \( \frac{dy}{dx} = -\frac{(3-x^{1/2})}{x^{1/2}} \) into the formula. This gives us:\[ 2\pi \int_{1}^{4} (3 - x^{1/2})^2 \sqrt{1 + \left( -\frac{(3-x^{1/2})}{x^{1/2}} \right)^2} \, dx \].

Step 4: Graph the Curve and Surface

Graph the original equation \( x^{1/2} + y^{1/2} = 3 \) to understand its shape between the points \((4,1)\) and \((1,4)\). Revolve this curve around the x-axis to visualize the surface of revolution. Use software or a graphing calculator to assist with the visualization.

Step 5: Evaluate the Integral Numerically

Use a graphing calculator or numerical integration tool to evaluate the integral for the surface area:\[ \text{Numerical Value} = \int_{1}^{4} (3 - x^{1/2})^2 \sqrt{1 + \left( \frac{(3-x^{1/2})}{x^{1/2}} \right)^2} \, dx \].The surface area is then \( 2\pi \times \text{Numerical Value} \).

Key Concepts

Integral Calculus

Integral calculus is a branch of mathematical analysis that deals with integration, a fundamental concept for calculating areas and volumes under and around curves. It involves finding a function's integral, which in many cases can represent the accumulation of quantities. In this context, integral calculus is used to determine the surface area generated by revolving a curve around an axis, a type of problem that frequently appears in physics and engineering applications.

• To find such surface areas, we often set up an integral that includes the original function and its derivative.
• This allows us to calculate the "accumulated" curved surface area over a specific interval.

In the given exercise, the problem is set up to calculate how much "surface" exists around the revolution of a curve described by \(x^{1/2} + y^{1/2} = 3\).

Chain Rule

The chain rule is a fundamental derivative rule in calculus, used to differentiate composite functions. When you have a function inside another, like in the exercise where \(y = (3 - x^{1/2})^2\), the chain rule helps find the derivative of these functions with respect to a variable. It is essential in determining how one small change in the variable impacts the function's value.

• Start by identifying the inner function and the outer function.
• For example, treat \(u = 3 - x^{1/2}\) and \(y = u^2\), helping to decompose the problem.
• Differentiate each part separately and then combine them as per the chain rule.

This rule was applied to find \(\frac{dy}{dx}\), which is valuable for solving our integral for the surface area.

Surface of Revolution

A surface of revolution is created when a curve is revolved around a line, often an axis like the x-axis. This results in a 3-dimensional shape whose surface area we might want to calculate, especially in applications like designing vessels or automobile parts.

• The typical approach involves defining a function that represents the curve.
• Use the formula \(2\pi \int_{a}^{b} y \sqrt{1+\left(\frac{dy}{dx}\right)^2} \, dx\) to set up the integral representing the surface area.
• Plug the function and its derivative into this integral to solve for the exact area.

In our example, after simplifying and differentiating the curve's function, the surface area of the revolved curve is found using these principles.

Numerical Integration

Numerical integration is a vital technique used when an integral is too complex to solve analytically, allowing us to approximate the integral's value. This is practical when solving real-world problems where exact functions might be less convenient to evaluate with traditional methods.

• Tools like graphing calculators or software apply numerical methods such as Simpson's Rule or the Trapezoidal Rule to approximate these values.
• In this case, after setting up the integral, you employ these tools to get the surface area numerically.
• This approach is integral for engineers and scientists who require quick and reliable results without delving into intricate details of every calculation.

The final numerical value from such integration gives us an approximation of the surface area of the revolution, crucial for practical applications.