Problem 46 - Calculus of a Single Variable

Question

Finding the Area of a Surface of Revolution In Exercises \(43-46,\) set up and evaluate the definite integral for the area of the surface generated by revolving the curve about the y-axis. $$ y=\frac{x}{2}+3, \quad 1 \leq x \leq 5 $$

Step-by-Step Solution

Answer

The surface area of the solid formed by revolving the curve \(y=\frac{x}{2}+3\), for \(1 \leq x \leq 5\), around the y-axis is 24\(\pi\sqrt{5}\) square units.

Step 1: Finding derivative dy/dx

The first step in the solution is differentiating the given function \(y = \frac{x}{2} + 3\) with respect to x, to get the derivative \(dy/dx = \frac{1}{2}\).

Step 2: Substituting dy/dx in the formula

Next, the derivative \(dy/dx = \frac{1}{2}\) will be substituted into the formula along the 'x' value. The integral formula becomes \(2\pi\int_{1}^{5}x\sqrt{1+ (\frac{1}{2})^2}dx\).

Step 3: Evaluating the Integral

Now evaluate the integral \(2\pi\int_{1}^{5}x\sqrt{1+ (\frac{1}{2})^2}dx = 2\pi\int_{1}^{5}\frac{x\sqrt{5}}{4}dx\). This simplifies to \(\frac{\pi\sqrt{5}}{2}\int_{1}^{5}xdx\) and on further simplification, the integral evaluates to 24\(\pi\sqrt{5}\).

Key Concepts

Definite Integral

The definite integral is a fundamental concept in calculus, representing the accumulation of quantities, such as areas under a curve, over an interval. In the context of finding the area of a surface of revolution, it helps us calculate the surface area formed when a curve is rotated around an axis.

When solving the problem, the definite integral is set up as part of the surface area formula. The curve given is revolved around the y-axis, and the integral is defined over the interval from 1 to 5. This means you are effectively measuring the accumulated area from when the curve begins at x = 1 until it ends at x = 5.

• The process involves identifying the appropriate integrand, which depends on the nature of the function being revolved and the axis of revolution.
• The definite integral helps in computing the precise value of the surface area, capturing the entire curved surface.

Understanding definite integrals is crucial as they translate real-world problems into solvable calculus equations. Mastery of this concept allows you to calculate a wide range of quantities beyond just areas, such as volumes and lengths of curves.

Derivative

The derivative is a tool that measures how a function changes as its input changes. It represents the slope of the function at any point and is fundamental to calculus.

In the context of surface of revolution problems like this one, finding the derivative is a critical first step. For the given function, \(y = \frac{x}{2} + 3\), the derivative \(\frac{dy}{dx}\) is calculated to be \(\frac{1}{2}\). This involves a straightforward application of basic differentiation techniques considering the linear nature of the function.

• The derivative is used to calculate the rate of change of the curve's height as \(x\) changes.
• It plays a key role in determining the element of arc length which is crucial for setting up the surface area integral.

Differentiation, in this context, helps us prepare the function before further integration steps. It ensures that when the curve is rotated, the change or "movement" of the curve is accurately maintained in our calculations.

Evaluating Integrals

Evaluating integrals involves calculating the exact value of an integral, which, in this problem, calculates the area of a surface formed by revolving a curve. After setting up the integral with the correct limits and integrand, as seen in the previous sections, evaluating the integral gives you the precise surface area.

The integral we need to evaluate is \(2\pi\int_{1}^{5}\frac{x\sqrt{5}}{4}dx\). This simplifies into easier parts: separating constants and dealing with the polynomial inside the integral.

• First, extract any outside constants from the integral for simplicity. Here, constants like \(\frac{\pi\sqrt{5}}{2}\) are separated.
• Next, integrate \(x\), which follows basic integration rules: \(\int x \, dx = \frac{x^2}{2}\).
• Substitute the limits after integrating to find the value: compute the result at the upper limit 5 and the lower limit 1 and subtract these values.

After performing these steps, you end up with the evaluated form, which yields \(24\pi\sqrt{5}\) in this example. This final value represents the total surface area of the revolution.