Question

$$ \begin{array}{l}{\text { Multiple Choice Which of the following expressions gives }} \\ {\text { the length of the graph of } x=y^{3} \text { from } y=-2 \text { to } y=2 ? \quad \mathrm{}}\end{array} $$ (A) \(\int_{-2}^{2}\left(1+y^{6}\right) d y \quad\) (B) \(\int_{-2}^{2} \sqrt{1+y^{6}} d y\) (C) \(\int_{-2}^{2} \sqrt{1+9 y^{4}} d y \quad\) (D) \(\int_{-2}^{2} \sqrt{1+x^{2}} d x\) $$ (\mathbf{E}) \int_{-2}^{2} \sqrt{1+x^{4}} d x $$

Short answer

The correct answer is (C) \(\int_{-2}^{2} \sqrt{1+9y^4} dy \).

Step-by-step solution

Identify and apply formula for length of curve

The length of a curve given by a function \( x=f(y) \) from \( y=a \) to \( y=b \) is given by the formula \( L = \int_{a}^{b} \sqrt{1+(f'(y))^2} dy \). Apply this formula to the function \( x=y^3 \).

Calculate the derivative

Differentiate \( x=y^3 \) with respect to \( y \), the derivative \( f'(y) \) is \( 3y^2 \).

Substitute the derivative in the formula

Substitute the derivative \( 3y^2 \) in the formula for \( L \). This gives \( L = \int_{a}^{b} \sqrt{1+(3y^2)^2} dy \). Simplify to get \( L = \int_{a}^{b} \sqrt{1+9y^4} dy \).

Identify the correct answer

Among the given options, the expression that matches the result obtained in Step 3 is option (C). Therefore, (C) is the answer.

Key concepts

Arc Length of a Curve

Understanding the arc length of a curve in calculus is essential when dealing with the geometry of curves. The arc length represents the distance along a curve between two points. In mathematical terms, for a smooth function represented as either y = f(x) or x = g(y), the arc length from a to b can be found using a specific integral formula.

For the curve x = g(y), the arc length formula is:
\[\begin{equation}\ L = \text{\int}_a^b \sqrt{1+(g'(y))^2} \text{dy} \end{equation}\]
where g'(y) is the derivative of g(y) with respect to y. This formula is derived from the Pythagorean Theorem, considering an infinitesimally small segment of the curve as the hypotenuse of a right triangle. The integral sums up these infinitesimal segments to calculate the total length of the curve between the two points.

In the context of the exercise, the curve given by x = y^3 requires the calculation of its length from y = -2 to y = 2. Following the steps of the solution, we can see how the formula is applied and the proper expression to obtain the arc length is identified.

It is crucial to correctly calculate the derivative and plug it into the arc length formula. Mistakes in this step can lead to the wrong integral expression for calculating the arc length.

Integrating Functions

Integrating functions is a fundamental concept in calculus, often required when solving for quantities such as area, volume, and, as in this exercise, arc length. The process of integration, the reverse of differentiation, involves finding a function whose derivative is the given function.

For calculating arc lengths, we integrate the square root of the sum of one and the square of the function's derivative:
\[\begin{equation}\ L = \text{\int}_a^b \sqrt{1+[f'(x)]^2} \text{dx} \end{equation}\]

or
\[\begin{equation}\ L = \text{\int}_a^b \sqrt{1+[g'(y)]^2} \text{dy} \end{equation}\]

depending on whether the function is in terms of x or y. When integrating, it's essential to consider the limits of integration which indicate the interval over which the integration is to be performed. Incorrect limits can result in an incorrect answer for the integral.

In the textbook exercise, integral option (C) represents the correct setup for the integral to find the arc length of the curve x = y^3. Recognizing the correct form of the integrand, as well as the correct limits of integration, is key to solving problems involving the integration of functions.

Calculus Problem Solving

Calculus problem solving involves a systematic approach to tackling mathematical challenges that require the knowledge of limits, derivatives, integrals, and the application of various theorems. Understanding the different techniques and when to apply them is critical.

Here's a simplified guide to the process:

• Understand the problem: Carefully read through the problem and identify what is being asked.
• Identify relevant formulas: Choose the appropriate formulas or theorems required to solve the problem.
• Perform algebraic manipulation: Make sure to simplify expressions when necessary for easier integration or differentiation.
• Carry out differentiation or integration: Apply the methods of calculus to find derivatives or integrals as per the demands of the problem.
• Interpret your result: Always relate your mathematical answer back to the context of the problem.

For instance, in the problem provided, it is essential to follow each step meticulously. Any error in differentiation, setting up the integral, or evaluating it can lead to an incorrect answer. In our example, the correct integration process led to the selection of option (C), thanks to correct differentiation and a clear understanding of the integral needed to calculate the arc length.

Effective problem solving in calculus also relies on recognizing patterns and using them to infer techniques or shortcuts that can simplify the overall process. Practicing a variety of problems increases familiarity and ease with the content, leading to improved problem-solving skills.