Question

Explain the steps required to find the length of a curve \(y=f(x)\) between \(x=a\) and \(x=b.\)

Short answer

Question: Find the length of the curve \(y=f(x)\) between \(x=a\) and \(x=b\) using the arc length formula in calculus. Explain each step in detail. Answer: To find the length of the curve between \(x=a\) and \(x=b\), follow these steps: 1. Understand the arc length formula: \(L = \int_{a}^{b}\sqrt{1+(\frac{dy}{dx})^2}dx\) 2. Find the derivative of the function, \(y=f(x)\): \(\frac{dy}{dx}\) 3. Square the derivative: \((\frac{dy}{dx})^2\) 4. Add 1 to the squared derivative: \(1+(\frac{dy}{dx})^2\) 5. Take the square root of the expression: \(\sqrt{1+(\frac{dy}{dx})^2}\) 6. Integrate the expression over the interval \([a, b]\): \(L = \int_{a}^{b}\sqrt{1+(\frac{dy}{dx})^2}dx\) 7. Simplify and find the length of the curve between \(x=a\) and \(x=b\).

Step-by-step solution

Understand the arc length formula

The arc length formula is derived from the Pythagorean theorem and is given by: $$L = \int_{a}^{b}\sqrt{1+(\frac{dy}{dx})^2}dx$$ L represents the length of the curve, and this formula integrates from \(x=a\) to \(x=b\).

Find the derivative of the function

To use the arc length formula, we need to find the derivative of the function \(y=f(x)\). The derivative, \(\frac{dy}{dx}\), represents the slope of the tangent line to the curve at any point. Find the derivative using the appropriate rules for the given function.

Square the derivative

Once we have found the derivative of the function, we need to square it. This is because the arc length formula uses the squared derivative: \((\frac{dy}{dx})^2\).

Add 1 to the squared derivative

Now, add 1 to the squared derivative: $$1+(\frac{dy}{dx})^2$$ This expression is found under the square root in the arc length formula.

Take the square root

Next, take the square root of the expression obtained in Step 4: $$\sqrt{1+(\frac{dy}{dx})^2}$$ This step simplifies the expression inside the integral.

Integrate the expression

Now that we have the simplified expression, integrate it over the interval \([a, b]\) to find the length of the curve: $$L = \int_{a}^{b}\sqrt{1+(\frac{dy}{dx})^2}dx$$ Use any relevant integration techniques required, and don't forget to evaluate the integral at the limits \(a\) and \(b\).

Simplify and find the length of the curve

Finally, simplify the result obtained after integrating and evaluating at the limits, and you'll get the length of the curve between \(x=a\) and \(x=b\).

Key concepts

Integration

Integration is a fundamental concept in calculus, critical for finding the arc length of a curve. It allows us to sum up infinitely small sections along the curve to compute its overall length. To integrate the expression from the arc length formula, which is \( L = \int_{a}^{b}\sqrt{1+(\frac{dy}{dx})^2}dx \), involves several steps.

• The expression under the square root, \( \sqrt{1+(\frac{dy}{dx})^2} \), needs to be continuous over the interval \([a, b]\) for the integration process to be applicable.
• Integration over a finite interval \([a, b]\) means finding the antiderivative and evaluating it at the boundaries \(x=a\) and \(x=b\). This step provides the exact arc length.
• Often, integration involves using methods like substitution or numerical techniques when an antiderivative is difficult to find analytically.

Integration is powerful because it not only helps compute areas under curves but also measures lengths, volumes, and more in various dimensions.
This concept ties back to easily measurable geometric ideas, making it crucial for solving numerous real-world problems.

Derivative

Derivatives play a key role in calculus and help us determine the rate of change at any point on a curve. When finding the arc length of a curve, the derivative \( \frac{dy}{dx} \) expresses the slope of the tangent at each point along the curve.

• To compute the derivative, different rules apply, such as the chain rule or the product rule, dependent upon the function \( f(x) \).
• Once calculated, the derivative helps us construct \( (\frac{dy}{dx})^2 \), which is integral in computing the arc length.
• The derivative gives insights into how a function behaves, whether it's increasing, decreasing, or reaching maximum/minimum values.

By understanding derivatives, you learn to "zoom in" on a curve and see how it turns and twists, information that is essential when determining the shape and length of that curve in greater detail.

Pythagorean Theorem

The Pythagorean Theorem is a classic principle in geometry, stating that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
This fundamental relationship lays the groundwork for the arc length formula, \( \sqrt{1+(\frac{dy}{dx})^2} \), which estimates the "hypotenuse" segmental lengths in small intervals along a curve.

• Each small piece of the curve can be viewed as a hypotenuse of a right triangle, formed by a small change in x and the corresponding change in y.
• By applying this theorem, we create the foundation for deriving many calculus concepts, like the arc length formula, marrying geometry with calculus.
• This theorem gaps the bridge between linear segments and more complex curved shapes.
  

As concepts in calculus, the Pythagorean Theorem helps extend geometric ideas into more profound mathematical tools like curves and surfaces, providing a base for computing lengths and areas.