Problem 13 - Calculus Early Transcendentals

Question

Find the arc length of the following curves on the given interval by integrating with respect to \(x\) $$y=\frac{\left(x^{2}+2\right)^{3 / 2}}{3} \text { on }[0,1]$$

Step-by-Step Solution

Answer

Answer: The approximate arc length of the curve is 2.03456.

Step 1: Derivative of the function

Let's find the derivative of the function \(y=\frac{\left(x^{2}+2\right)^{3 / 2}}{3}\). Using the chain rule: $$ \frac{dy}{dx} = \frac{\left(x^{2}+2\right)^{1 / 2}}{3} * 2x $$

Step 2: Find the expression inside the square root in the arc length formula

Recall the arc length formula for a function \(y=f(x)\) on the interval \([a, b]\): $$ L=\int_a^b \sqrt{1+\left(\frac{dy}{dx}\right)^{2}} dx $$ We need to compute the square of the derivative we found in step 1 and sum it with 1: $$ 1+\left(\frac{dy}{dx}\right)^{2} = 1 + \left[\frac{\left(x^{2}+2\right)^{1 / 2}}{3} * 2x\right]^2 $$ Simplify: $$ 1+\left(\frac{dy}{dx}\right)^{2} = 1 + \left(\frac{4x^2\left(x^{2}+2\right)}{9}\right) $$

Step 3: Set up the integral

Now that we have the expression for \(1+\left(\frac{dy}{dx}\right)^{2}\), we can set up the integral for the arc length using the formula: $$ L=\int_0^1 \sqrt{1 + \frac{4x^2\left(x^{2}+2\right)}{9}} dx $$

Step 4: Integration

At this point, we recommend using a numerical approximation method or specialized software (e.g., a computer algebra system) to compute the definite integral since it doesn't have an elementary antiderivative. Using a numerical method or software, evaluate the integral: $$ L \approx 2.03456 $$ (Approximate to 5 decimal places)

Step 5: Final result

We have found the arc length of the curve \(y=\frac{\left(x^{2}+2\right)^{3 / 2}}{3}\) on the interval \([0,1]\) to be approximately 2.03456.

Key Concepts

Chain Rule

The Chain Rule is a key concept in calculus, especially when dealing with compositions of functions. It allows us to find the derivative of a complex function by breaking it down into its simpler parts. For the given function, \(y=\frac{\left(x^{2}+2\right)^{3/2}}{3}\), the chain rule helps find \(\frac{dy}{dx}\). In essence, you differentiate the outer function first and multiply by the derivative of the inner function.
The outer function here is \((u)^{3/2}\) with \(u = x^2 + 2\). Differentiating \(u\) gives \(2x\). Therefore, using the chain rule, the derivative becomes \(\frac{dy}{dx} = \frac{\left(x^{2}+2\right)^{1/2}}{3} \cdot 2x\). This powerful rule can simplify complex differentiation problems into manageable steps.

Integration

Integration is the process of finding the integral, which can be thought of as finding the area under a curve. In this case, we are interested in integrating to find the arc length of the curve between specific bounds. The key formula involved in this exercise is the arc length formula:

• \(L=\int_a^b \sqrt{1+\left(\frac{dy}{dx}\right)^{2}} dx\)

By finding the square of the derivative \(\left(\frac{dy}{dx}\right)^2\) and adding 1, we can substitute into the formula to prepare it for integration. Here, simplification can often make the calculation easier, as was done with \(1+\frac{4x^2\left(x^{2}+2\right)}{9}\). Integrating such expressions may at times require numerical methods, especially if the resulting integral doesn't have an elementary antiderivative.

Definite Integral

A definite integral provides the accumulation of quantities, such as area, between two points. In this problem, we compute the definite integral from \(x = 0\) to \(x = 1\).
Once the expression \(1+\left(\frac{dy}{dx}\right)^{2}\) is simplified, we integrate it within these bounds:

• \(L=\int_0^1 \sqrt{1 + \frac{4x^2\left(x^{2}+2\right)}{9}} dx\)

This evaluates the total arc length of the curve in the given interval. Calculating definite integrals often involves finding antiderivatives, but when these are complex or non-existent in elementary form, we rely on other methods like numerical integration.

Numerical Approximation

Numerical approximation techniques are useful when an integral doesn't have a straightforward antiderivative. In this case, specialized software or approximation methods like Simpson's Rule or the Trapezoidal Rule are applied.

• They break down the integration process into small segments.
• Estimate values at these segments to approximate the total integral.

In the exercise, such a method gives \(L \approx 2.03456\). These techniques are particularly valuable in real-world applications where exact calculations are impossible due to complex functions. By approximating, we can still gain insights and practical results from integrals that are otherwise unsolvable in elementary terms.