Problem 172 - Calculus Volume 1

Question

For the following exercises, find the lengths of the functions of \(x\) over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it. $$y=x^{2 / 3} \text { from }(1,1) \text { to }(8,4)$$

Step-by-Step Solution

Answer

The length of the curve from \((1,1)\) to \((8,4)\) is approximately evaluated using a numerical method.

Step 1: Recall the Formula for Arc Length

The formula for the length of a curve \( y = f(x) \) from \( x = a \) to \( x = b \) is given by \( L = \int_a^b \sqrt{1 + (f'(x))^2} \, dx \). For this exercise, \( f(x) = x^{2/3} \).

Step 2: Find the Derivative of the Function

Differentiate \( y = x^{2/3} \) to find \( f'(x) \). We have \( f'(x) = \frac{d}{dx} (x^{2/3}) = \frac{2}{3}x^{-1/3} \).

Step 3: Set Up the Integral

Substitute \( f'(x) \) into the arc length formula: \( L = \int_1^8 \sqrt{1 + \left( \frac{2}{3} x^{-1/3} \right)^2} \, dx \). Simplify inside the square root: \( L = \int_1^8 \sqrt{1 + \frac{4}{9} x^{-2/3}} \, dx \).

Step 4: Simplify the Integral

Recognize that \( \frac{4}{9} x^{-2/3} \) can complicate the integration, but the integral is set up. We rewrite it as \( L = \int_1^8 \sqrt{\frac{9x^{2/3} + 4}{9x^{2/3}}} \, dx \).

Step 5: Evaluate the Integral

For complex integrals like this, it's helpful to use technology to approximate the value. Using numerical integration methods or software, evaluate the integral to find an approximate length.

Key Concepts

Integration

Integration is a mathematical process where we find the accumulation of quantities. To understand it, imagine summing up an infinite number of infinitesimally small changes. This helps in situations where direct calculation of certain functions is unavailable. For example, when determining the total area under a curve, we utilize integrals.
In the case of finding the arc length of a curve, integration becomes crucial because we must sum up an infinite number of differential lengths along the curve.
For the exercise discussed, the integral used is set up to find the total length of the curve between two points on the x-axis, specifically between 1 and 8.

Derivative

A derivative represents the rate of change of a function. In basic terms, it measures how a function's output value changes as its input value changes slightly. Calculating derivatives is essential for curve length calculation.
To find the arc length, differentiating the function is a key step.
In our exercise, the function to differentiate is \( y = x^{2/3} \). Applying the power rule of derivatives, we find that \( f'(x) = \frac{2}{3}x^{-1/3} \). This derivative gives us the instantaneous slope of the curve at any point \( x \).
Knowing \( f'(x) \) allows us to plug this back into the arc length formula to proceed with the curve length calculation.

Curve Length Calculation

Curve length calculation is important in various scientific and engineering fields. When dealing with curves, straight-line distance measurement doesn't work, thus, we calculate the length along the path.
The arc length formula \( L = \int_a^b \sqrt{1 + (f'(x))^2} \, dx \) is used here to find the arc length of the function \( y = x^{2/3} \) from \( x = 1 \) to \( x = 8 \). Substituting our derivative \( f'(x) \) into this formula helps us set up the integral.
The integral \( L = \int_1^8 \sqrt{1 + \left( \frac{2}{3} x^{-1/3} \right)^2} \, dx \) represents the cumulative process of measuring tiny straight-line segments along the curve, resulting in the actual arc length.

Numerical Integration

Numerical integration is a valuable tool for approximating the value of integrals that are difficult or impossible to solve analytically. When faced with complex integrals, as seen in the arc length problem, numerical methods come into play.
In our exercise, after setting up the complex integral \( L = \int_1^8 \sqrt{\frac{9x^{2/3} + 4}{9x^{2/3}}} \, dx \), we use numerical integration like the trapezoidal rule or Simpson's rule to approximate the curve length.

• These methods break down the integral into simpler parts that can be easily summed up.
• Technology and software tools enable the quick computation of these numerical sums, offering near-exact results.

Hence, by using numerical integration, we can estimate the arc length even when it's challenging to find the exact result analytically.