Problem 3 - Essential Calculus

Question

In Exercises \(3-14,\) find the arc length of the graph of the function over the indicated interval. $$ y=\frac{2}{3} x^{3 / 2}+1, \quad[0,1] $$

Step-by-Step Solution

Answer

The arc length of the graph of the function \( y = \frac{2}{3} x^{3 / 2} + 1 \) on the interval \([0, 1]\) is \( L = \frac{2}{3}(2\sqrt{2} - 1) \).

Step 1: Compute the derivative

Find the derivative \( f'(x) \) of the function \( y = f(x) = \frac{2}{3} x^{3 / 2} + 1\). The derivative is \( f'(x) = (2/3) \cdot (3/2)x^{1/2} = x^{1/2}\).

Step 2: Plug the derivative into the arc length formula

Substitute \( f'(x)\) into the integral that represents the arc length formula which is \( \int_a^b \sqrt{1+[f'(x)]^2} dx\). This gives us: \( L = \int_0^1 \sqrt{1 + [(x^{1/2})]^2} dx = \int_0^1 \sqrt{1 + x} dx\).

Step 3: Evaluate the integral

This integral can be evaluated by the method of substitution. Let \( u = x +1 \), then \( dx = du \) and when \( x = 0, u = 1 \), and when \( x = 1, u = 2 \). So the integral becomes \( L = \int_1^2 \sqrt{u} du \). Integrating, we get \( L = [\frac{2}{3}u^{3/2}]_1^2 = \frac{2}{3}(2^{3/2} - 1) = \frac{2}{3}(2\sqrt{2} - 1)\).

Key Concepts

Derivative

The derivative is a fundamental concept in calculus that measures the rate at which a function changes. For the given function \( y = \frac{2}{3} x^{3/2} + 1 \), we need to find \( f'(x) \). To obtain the derivative, apply the power rule, which states that if \( y = x^n \), then \( f'(x) = n x^{n-1} \). Here, the term \( x^{3/2} \) has a derivative of \( (3/2)x^{1/2} \) when applying the power rule. Multiply by the constant \( \frac{2}{3} \) to get \( f'(x) = x^{1/2} \). This step is essential for further calculations like finding the arc length.

Integral

Integrals are used to calculate accumulated quantities, such as areas under curves or, in this case, the arc length of a curve. Once we have the derivative, the next step is to set up the integral to find the arc length. The formula for arc length \( L \) is \( L = \int_a^b \sqrt{1+[f'(x)]^2} \, dx \). We substitute \( f'(x) = x^{1/2} \) into the formula, resulting in \( \int_0^1 \sqrt{1 + x} \, dx \). The integral captures the gradual change in length as you move along the curve, helping us to compute the total arc length from \( x=0 \) to \( x=1 \).

Substitution Method

The substitution method is a technique used to simplify integrals, making them easier to evaluate. It's particularly useful when dealing with composite functions or those that are difficult to integrate directly. In this problem, the integral \( \int_0^1 \sqrt{1 + x} \, dx \) is evaluated using substitution.

• Let \( u = x + 1 \), then \( du = dx \).
• When \( x = 0 \), \( u \) becomes 1.
• When \( x = 1 \), \( u \) becomes 2.

This substitution simplifies our integral to \( \int_1^2 \sqrt{u} \, du \), which is easier to solve. Such transformation allows for computations with complex functions by changing them into simpler forms. It's a very helpful tool in the calculus toolbox.

Calculus Problem Solving

Successfully solving calculus problems involves a clear understanding of the principles and methods involved, like differentiation and integration. In this specific problem, we're tasked to find the arc length of a function, which is a typical calculus application. The steps followed:

• Compute the derivative of the function to plug into the arc length formula.
• Set up the integral required for the arc length calculation.
• Apply the substitution method to simplify and evaluate the integral.

Approaching problems methodically and employing these techniques allows for comprehensive solutions. It's vital to practice these processes to hone problem-solving skills and understand the underlying concepts.