Question

The length of \(y\) for \(x=3-\sqrt{y}\) from \(y=0\) to \(y=4\)

Short answer

The arc length from \( y = 0 \) to \( y = 4 \) on the curve is calculated using integration.

Step-by-step solution

Express x in terms of y

The given function is expressed as \( x = 3 - \sqrt{y} \). Rearrange this formula to express \( \sqrt{y} \) in terms of \( x \), giving \( \sqrt{y} = 3 - x \). By squaring both sides, \( y = (3 - x)^2 \).

Determine limits for x

Find the range for \( x \) by using the given range for \( y \) which is from 0 to 4. \( y = 0 \) gives \( x = 3 - \sqrt{0} = 3 \) and \( y = 4 \) gives \( x = 3 - \sqrt{4} = 1 \). So, \( x \) ranges from 1 to 3.

Set up the arc length formula

The formula for arc length \( L \) of a curve \( y = f(x) \) is \( L = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2 } \, dx \). Substitute \( y = (3 - x)^2 \) in place of \( y \), with limits \( x = 1 \) to \( x = 3 \).

Find derivative dy/dx

Differentiate \( y = (3 - x)^2 \) with respect to \( x \). Using the chain rule, \( \frac{dy}{dx} = 2(3-x)(-1) = -2(3-x) \). Thus, \( \frac{dy}{dx} = -2(3-x) \).

Substitute derivative into the arc length formula

Substitute \( \frac{dy}{dx} \) into the arc length formula: \[ L = \int_{1}^{3} \sqrt{1 + [-2(3-x)]^2 } \, dx \] This simplifies to \[ L = \int_{1}^{3} \sqrt{1 + 4(3-x)^2} \, dx \].

Calculate the definite integral

Solve the integral \[ L = \int_{1}^{3} \sqrt{1 + 4(3-x)^2} \, dx \]. First simplify inside: \( 1 + 4(3-x)^2 = 4(x - 3)^2 + 1 \).This integral requires a substitution or numerical methods.Approximate numerically, or use known methods if specific software is allowed.

Key concepts

Integration

Integration is a fundamental concept in calculus, used to find the total accumulation of quantities. In the context of arc length, we use integration to calculate the length of a curve over a specified interval. This involves evaluating a definite integral, which includes specific limits of integration.
For the arc length of our function, the formula is:

• \[L = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2 } \, dx\]

The integral takes into account the entire curve from the beginning of our limit, \( a \), to the end, \( b \). By substituting and simplifying the expression inside the square root, we can find the arc length for virtually any differentiable function.

Differentiation

Differentiation allows us to compute the rate at which one quantity changes with respect to another. In our problem, we differentiate the curve equation to find the slope at each point, which is necessary in calculating the arc length.
The process involves using basic rules of differentiation, including the power rule and the chain rule, to find \( \frac{dy}{dx} \). For example:

• Starting from \( y = (3 - x)^2 \), differentiate to find the derivative.
• Using the chain rule, we have: \[\frac{dy}{dx} = -2(3-x)\]

Differentiation turns our curve formula into an expression representing how vertically steep the curve is at any point. This is crucial for the arc length formula, where \( \left( \frac{dy}{dx} \right)^2 \) appears under a square root as part of the integrand.

Limits of Integration

The limits of integration define the interval over which we evaluate the integral. These limits are essential for calculating quantities like area under a curve or arc length.
In this exercise, we find the limits of integration by following these steps:

• Identify the given range for \( y \), which is from 0 to 4.
• Solve for corresponding \( x \) values using the relationship \( x = 3 - \sqrt{y} \).
• Find that when \( y = 0 \), \( x = 3 \), and when \( y = 4 \), \( x = 1 \).

Thus, our limits of integration for calculating the arc length of the curve are from 1 to 3. These define the section of the curve over which the arc length is measured, ensuring the calculation is accurate and bounded to the described segment of the function.

Chain Rule

The chain rule is a basic technique in calculus used to differentiate composite functions. It is indispensable when dealing with functions nested within one another, such as \( y = (3-x)^2 \).
To apply the chain rule:

• Identify the outer and inner functions. Here, the outer function is \( u^2 \), where \( u = 3-x \).
• Differentiate the outer function, keeping the inner function as is, and then multiply by the derivative of the inner function: \[\frac{d}{dx}(u^2) = 2u \cdot \frac{d}{dx}(3-x)\]
• Substituting, we get: \[\frac{dy}{dx} = -2(3-x)\]

The chain rule simplifies situations where functions "wrap" one another, enabling the differentiation of complex equations piece by piece. It is especially useful for scenarios like calculating arc lengths, where precise derivatives of composite functions are needed.