Arc Length Calculation Of F(y)=(y+1)^(3/2) From (8,3) To (27,8)

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The arc length of a curve is a fundamental concept in calculus that allows us to determine the distance along a curved path. In this article, we will delve into the process of calculating the arc length of a specific function, $f(y) = (y+1)^{\frac{3}{2}}$, between the points (8,3) and (27,8). This problem provides a practical application of integral calculus and highlights the importance of understanding curve parametrization and the arc length formula.

Understanding Arc Length

Before we dive into the specifics of our problem, let's establish a solid understanding of what arc length represents. Imagine a curve drawn in a two-dimensional plane. The arc length is simply the distance you would travel if you were to walk along that curve from one point to another. It's analogous to measuring the length of a piece of string that perfectly traces the curve.

Mathematically, the arc length of a curve defined by a function y = f(x) from x = a to x = b is given by the following integral:

$\qquad L = \int_{a}^{b} \sqrt{1 + [f'(x)]^2} \, dx$

This formula arises from the Pythagorean theorem, where we consider infinitesimally small segments of the curve and approximate their lengths using the square root of the sum of the squares of the horizontal and vertical changes. Summing up these infinitesimal lengths over the interval [a, b] gives us the total arc length.

In our case, the function is given in terms of y, so we will use a slightly modified version of the formula:

$\qquad L = \int_{c}^{d} \sqrt{1 + [f'(y)]^2} \, dy$

where f(y) is the function, and we are integrating with respect to y from y = c to y = d. This adjustment is necessary because our curve is defined as a function of y rather than x. The points (8,3) and (27,8) provide the limits of integration in terms of the y-values, which are 3 and 8, respectively.

Problem Setup: Arc Length of $f(y)=(y+1)^{\frac{3}{2}}$

Now, let's focus on the specific problem at hand. We are tasked with finding the arc length of the function $f(y) = (y+1)^{\frac{3}{2}}$ from the point (8,3) to the point (27,8). This means we need to determine the distance along the curve defined by this function between these two points.

The first crucial step is to identify the limits of integration. Since our function is defined in terms of y, we need the y-coordinates of the given points. These are 3 and 8, corresponding to the points (8,3) and (27,8), respectively. Therefore, our integration will be performed from y = 3 to y = 8.

Next, we need to find the derivative of the function $f(y)$ with respect to y. This derivative, denoted as $f'(y)$, represents the instantaneous rate of change of the function and is essential for calculating the arc length. Once we have the derivative, we will substitute it into the arc length formula and evaluate the resulting integral.

Calculating the Derivative $f'(y)$

To find the derivative of $f(y) = (y+1)^{\frac{3}{2}}$, we will use the power rule and the chain rule of differentiation. The power rule states that the derivative of $x^n$ is $nx^{n-1}$, while the chain rule is used when differentiating a composite function. Applying these rules, we have:

$\qquad f(y) = (y+1)^{\frac{3}{2}}$

$\qquad f'(y) = \frac{3}{2}(y+1)^{\frac{3}{2} - 1} \cdot \frac{d}{dy}(y+1)$

$\qquad f'(y) = \frac{3}{2}(y+1)^{\frac{1}{2}} \cdot 1$

$\qquad f'(y) = \frac{3}{2}\sqrt{y+1}$

Thus, the derivative of our function is $f'(y) = \frac{3}{2}\sqrt{y+1}$. This derivative will be used in the arc length formula to calculate the length of the curve between the specified points. The derivative represents the slope of the tangent line to the curve at any given point, and it plays a crucial role in determining the arc length.

Applying the Arc Length Formula

Now that we have the derivative, $f'(y) = \frac{3}{2}\sqrt{y+1}$, we can substitute it into the arc length formula:

$\qquad L = \int_{c}^{d} \sqrt{1 + [f'(y)]^2} \, dy$

In our case, c = 3 and d = 8, so the integral becomes:

$\qquad L = \int_{3}^{8} \sqrt{1 + \left[\frac{3}{2}\sqrt{y+1}\right]^2} \, dy$

Let's simplify the expression inside the square root:

$\qquad L = \int_{3}^{8} \sqrt{1 + \frac{9}{4}(y+1)} \, dy$

$\qquad L = \int_{3}^{8} \sqrt{1 + \frac{9}{4}y + \frac{9}{4}} \, dy$

$\qquad L = \int_{3}^{8} \sqrt{\frac{4}{4} + \frac{9}{4}y + \frac{9}{4}} \, dy$

$\qquad L = \int_{3}^{8} \sqrt{\frac{9}{4}y + \frac{13}{4}} \, dy$

$\qquad L = \int_{3}^{8} \sqrt{\frac{1}{4}(9y + 13)} \, dy$

$\qquad L = \int_{3}^{8} \frac{1}{2}\sqrt{9y + 13} \, dy$

Now we have a more manageable integral to evaluate. This integral represents the arc length of the curve $f(y) = (y+1)^{\frac{3}{2}}$ from y = 3 to y = 8. The next step is to solve this integral, which will give us the numerical value of the arc length.

Evaluating the Integral

To evaluate the integral, we can use a u-substitution. Let:

$\qquad u = 9y + 13$

Then,

$\qquad du = 9 \, dy$

$\qquad dy = \frac{1}{9} \, du$

We also need to change the limits of integration. When y = 3:

$\qquad u = 9(3) + 13 = 27 + 13 = 40$

When y = 8:

$\qquad u = 9(8) + 13 = 72 + 13 = 85$

Now we can rewrite the integral in terms of u:

$\qquad L = \frac{1}{2} \int_{40}^{85} \sqrt{u} \cdot \frac{1}{9} \, du$

$\qquad L = \frac{1}{18} \int_{40}^{85} u^{\frac{1}{2}} \, du$

Now we can integrate using the power rule for integration:

$\qquad L = \frac{1}{18} \left[ \frac{2}{3}u^{\frac{3}{2}} \right]_{40}^{85}$

$\qquad L = \frac{1}{27} \left[ u^{\frac{3}{2}} \right]_{40}^{85}$

Now we evaluate the expression at the limits of integration:

$\qquad L = \frac{1}{27} \left[ (85)^{\frac{3}{2}} - (40)^{\frac{3}{2}} \right]$

Calculating the Numerical Value

Using a calculator, we find the numerical values:

$\qquad (85)^{\frac{3}{2}} \approx 783.3246$

$\qquad (40)^{\frac{3}{2}} \approx 252.9822$

Substituting these values back into the expression for L:

$\qquad L = \frac{1}{27} \left[ 783.3246 - 252.9822 \right]$

$\qquad L = \frac{1}{27} \left[ 530.3424 \right]$

$\qquad L \approx 19.6423$

Rounding to the nearest thousandth, we get:

$\qquad L \approx 19.642$

Conclusion

Therefore, the arc length of the graph of $f(y) = (y+1)^{\frac{3}{2}}$ from (8,3) to (27,8) is approximately 19.642 units. This problem demonstrates the application of integral calculus to find the length of a curve, highlighting the importance of understanding derivatives, the arc length formula, and techniques of integration such as u-substitution. The process involves careful calculation and attention to detail, but it provides a powerful tool for measuring distances along curved paths.

Arc Length = 19.642