Question

Multiple Choice Which of the following expressions should be used to find the length of the curve \(y=x^{2 / 3}\) from \(x=-1\) to \(x=1 ?\) \((\mathbf{A}) 2 \int_{0}^{1} \sqrt{1+\frac{9}{4} y} d y \quad\) (B) \(\int_{-1}^{1} \sqrt{1+\frac{9}{4} y} d y\) (C) \(\int_{0}^{1} \sqrt{1+y^{3}} d y \quad\) (D) \(\int_{0}^{1} \sqrt{1+y^{6}} d y\) \((\mathbf{E}) \int_{0}^{1} \sqrt{1+y^{9 / 4}} d y\)

Short answer

None of the options provided in the exercise match the correct integral for finding the length of the curve.

Step-by-step solution

Find the Derivative

The first step is to find the derivative \(f'(x)\) of the function \(y = x^{2/3}\). The power rule for differentiation gives \(f'(x) = (2/3)x^{-1/3}\)

Calculate the Expression under the Square Root

Next, calculate the expression under the square root in the arc length formula: \(1 + [f'(x)]^2 = 1 + [(2/3)x^{-1/3}]^2 = 1 + (4/9)x^{-2/3}\)

Identify the Limits of Integration and the Correct Integral

Finally, since we want the length of the curve from \(x = -1\) to \(x = 1\), we compare our integral to the given options. The correct integral is \(\int_{-1}^{1} \sqrt{1 + (4/9)x^{-2/3}} dx\). None of the options match this integral perfectly, so the answer comes down to how they have been simplified.

Key concepts

Derivative Calculation

When dealing with calculus problems, especially those concerning arc lengths, the derivative calculation is the first critical step. The derivative gives us a function's rate of change at any given point. In our example, we have the function y = x^{2/3}, and we need its derivative to proceed with the arc length calculation.

The power rule, a fundamental concept in calculus, is used to find derivatives of functions like y = x^n where n is any real number. According to the power rule, the derivative of x^n is nx^{(n-1)}. Applying this rule, the derivative of y = x^{2/3} is found to be (2/3)x^{-1/3}. It's important to remember the negative exponent which indicates division by that power of x - something that can be easily overlooked but is crucial for correct calculations.

Arc Length Formula

The arc length formula is key to finding the distance along a curve and is an important application of integral calculus. For a function y = f(x) defined on a closed interval [a, b], the arc length L is given by L = \[ \[\int_{a}^{b} \sqrt{1 + [f'(x)]^2} dx\] \], where f'(x) is the derivative of f(x).

In the step-by-step solution, after finding the derivative f'(x), we substitute it into the formula and square it. It's vital to square the entire derivative, not just the coefficient, a mistake some students might make. The integrand, therefore, includes 1 + (4/9)x^{-2/3} inside the square root, which simplifies the process of finding the arc length. Notice that this integrand captures not just the slope at each point (given by f'(x)), but also accounts for the inherent 'steepness' of the curve by squaring the derivative.

Limits of Integration

Limits of integration define the interval over which we integrate, and selecting the correct limits is critical for accurate results. In arc length problems, limits of integration correspond to the segment of the curve we're interested in. For the curve y = x^{2/3}, we're asked to find the arc length from x = -1 to x = 1. These points serve as our limits of integration.

Integrating within these limits accounts for the entire stretch of the curve we're examining. The reason we don't see direct options matching our integral in the solution is often due to simplification or a re-expression of the function. As students work through these problems, they need to ensure their limits of integration align with the context of the question. In some cases, they may need to consider symmetry or other properties of the curve that could simplify the integral. It’s also important to recognize when an improper integral is needed, which happens if the function approaches infinity, or the integration limits include a point of discontinuity.