Question

Find the length of \(y=x^{3 / 2}\) between \(x=0\) and \(x=5\)

Short answer

Question: Calculate the length of the curve \(y = x^{3/2}\) from \(x = 0\) to \(x = 5\). Answer: The length of the curve \(y = x^{3/2}\) from \(x = 0\) to \(x = 5\) is approximately \(19.028\).

Step-by-step solution

Find the derivative of the function

To find the first derivative of the function \(y = x^{3/2}\), we will use the power rule for derivatives. The power rule states that if \(y = x^n\), then \(y' = nx^{n-1}\). Applying the power rule, we have: $$y' = \frac{3}{2}x^{\frac{3}{2} - 1} = \frac{3}{2}x^{1/2}$$

Plug the derivative into the arc length formula

Now that we have the derivative, we will plug it into the arc length formula: $$L = \int_a^b \sqrt{1 + [f'(x)]^2} dx = \int_0^5 \sqrt{1 + [\frac{3}{2}x^{1/2}]^2} dx$$

Simplify the integral

In this step, we will simplify the integral before solving it: $$L = \int_0^5 \sqrt{1 + (\frac{9}{4}x)} dx$$

Solve the integral

To solve the integral, we will use the substitution method. Let \(u = 1 + \frac{9}{4}x\), and then \(du = \frac{9}{4}dx\). Since \(x = 0\), we get the lower limit for \(u\) to be \(u(0) = 1\), and for \(x = 5\), we get the upper limit for \(u\) to be \(u(5) = 1 + \frac{9}{4}(5) = \frac{41}{4}\). So, we can rewrite our integral in terms of \(u\) as: $$L = \int_1^{\frac{41}{4}} \sqrt{u} \cdot \frac{4}{9}du = \frac{4}{9}\int_1^{\frac{41}{4}} u^{1/2} du$$ Now, we can use the power rule for integration: $$L = \frac{4}{9}(2u^{3/2})\Big|_1^{\frac{41}{4}} = \frac{8}{9}\left[\left(\frac{41}{4}\right)^{3/2} - 1\right]$$

Evaluate the arc length

Finally, we will evaluate the arc length as follows: $$L = \frac{8}{9}\left[\left(\frac{41}{4}\right)^{3/2} - 1\right] \approx 19.028$$ Thus, the length of the curve \(y = x^{3/2}\) between \(x = 0\) and \(x = 5\) is approximately \(19.028\).

Key concepts

Power Rule for Derivatives

When it comes to differentiating functions, one of the most basic and widely used techniques is the power rule. This rule is particularly helpful for functions where the variable, usually denoted as x, is raised to a power, written as x^n. The power rule simplifies differentiation by allowing us to bring down the exponent as a coefficient and then subtract one from the exponent.

The formula for the power rule is quite straightforward: if y = x^n, then the derivative of y with respect to x, denoted as y', is y' = n*x^(n-1). This rule saves time and reduces the complexity of calculations, particularly when dealing with higher powers of x.

In our exercise, for the function y = x^(3/2), applying the power rule gives us y' = (3/2)*x^(1/2), laying the groundwork for further steps in finding the arc length.

Integral Calculus

Integral calculus is akin to piecing together a puzzle. It revolves around finding the whole when you only have parts, and it is central to many applications in physics and engineering, such as computing areas under curves and volumes of solids. The process of integration takes a function and returns the area underneath its curve from a boundary a to b.

To perform an integration, you usually need a continuous function and its interval. In our example, we are trying to find the arc length, which requires us to integrate over the length of the curve. The integral we are dealing with is not just of the function itself, but of the square root of 1 plus the function's derivative squared, as prescribed by the arc length formula.

Substitution Method in Integration

The substitution method in integration, also known as u-substitution, is a technique used to simplify integrals that might appear complex at first glance. This method involves choosing a part of the integral to replace with a variable, usually u, making the integral easier to solve.

It's comparable to a change of variables, providing a new perspective from which the integral might be more straightforward. Our exercise demonstrates this method perfectly. We identify u = 1 + (9/4)x, and the derivative of u with respect to x, du, is (9/4)dx, which we then solve for dx. This transformation allows us to rewrite the integral in a simpler form and use the power rule for integration to find the arc length.

Arc Length Formula

The arc length of a curve is the distance along the curve from one point to another. Calculating it involves integral calculus, specifically a formula that combines the derivatives of a function and the limits of integration. The arc length formula for a function y = f(x) between two points a and b on the x-axis is: L = \( \int_a^b \sqrt{1 + [f'(x)]^2} dx \).

This formula may seem daunting, but it is essentially the integration of the square root of 1 plus the square of the derivative of the function over the desired interval. In our example, after finding the derivative of the function using the power rule for derivatives and plugging it into the arc length formula, we are able to integrate to find the precise length of the curve.