Question

Figure \(5.1\) shows the portion of the curve \(y(x)\) between \(x=a\) and \(x=b\). A small piece of this curve has been selected and can be considered as the hypotenuse of a triangle with base \(\delta x\) and height \(\delta y\). (a) Use Pythagoras's theorem to find the length of the hypotenuse. (b) By summing all such contributions between \(x=a\) and \(x=b\), and letting \(\delta x \rightarrow 0\), obtain the integral expression for the total length of the curve.

Short answer

Answer: The integral expression for the total length of the curve y(x) between x=a and x=b is given by: \(\int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} dx\).

Step-by-step solution

Find the length of a small piece of the curve using Pythagoras's theorem

Consider a small piece of the curve y(x) with base length \(\delta x\) and height length \(\delta y\). This small piece can be thought of as the hypotenuse of a right triangle. We can use Pythagoras's theorem to find the length of this hypotenuse: Hypotenuse length = \(\sqrt{(\delta x)^2 + (\delta y)^2}\)

Finding an expression for \(\delta y\) in terms of \(\delta x\) using the derivative of y(x)

The slope of the tangent line at any point on the curve is given by the derivative of y(x). Thus, we can write: Slope = \(\frac{\delta y}{\delta x} = \frac{dy}{dx}\) Rearranging this expression, we have: \(\delta y = \frac{dy}{dx} \delta x\)

Substitute the expression for \(\delta y\) into the equation for the hypotenuse length

Replacing \(\delta y\) in the equation for the length of the hypotenuse found in Step 1 with the expression found in Step 2: Hypotenuse length =\(\sqrt{(\delta x)^2 + \left(\frac{dy}{dx}\delta x\right)^2}\)

Simplify the hypotenuse length expression

Factor out \((\delta x)^2\) from the hypotenuse length expression: Hypotenuse length =\(\delta x \sqrt{1 + \left(\frac{dy}{dx}\right)^2}\)

Obtain the integral expression for the total length of the curve

To find the total length of the curve, we need to sum over all the small hypotenuse lengths between x=a and x=b. We will do this by replacing the summation with an integral and taking the limit as \(\delta x \rightarrow 0\). This gives us the integral expression for the total length of the curve: Total length = \(\int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} dx\) Thus, the integral expression for the total length of the curve y(x) between x=a and x=b is given by: \(\int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} dx\)

Key concepts

Arc Length

When we explore the concept of arc length in calculus, we're trying to find the length of a curve between two specific points. This involves looking at the curve almost like a collection of tiny straight lines, which helps us calculate the total distance along the curve. The magic of calculus allows us to accurately measure this—even for curves that twist and turn.
To determine the arc length, we break the curve into small segments, each resembling the hypotenuse of a tiny triangle. Then, using Pythagorean theorem, we can calculate the length of each segment. Summing up the length of these segments gives us the arc length for the entire curve.
The final step involves an integral, as this allows us to consider all the infinitesimal pieces of the curve, from start to end points. The integral expression for the arc length between points \(x=a\) and \(x=b\) is:\[\int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx\]This formula takes the sum of all those tiny hypotenuses along the curve—a process only possible through the powers of integral calculus.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry. It describes the relationship between the sides of a right triangle. If \(a\) and \(b\) are the lengths of the legs and \(c\) is the length of the hypotenuse, then the theorem is expressed as:\[ a^2 + b^2 = c^2 \]In the context of finding the arc length of a curve, we apply this theorem on a very tiny scale. We imagine each tiny segment of the curve as forming a right triangle. The base of this triangle is a small change in \(x\), noted as \(\delta x\), and the height is a small change in \(y\), noted as \(\delta y\).
By applying Pythagoras's theorem, the hypotenuse—which represents the length of this curve segment—is:\[\sqrt{(\delta x)^2 + (\delta y)^2}\]This step-by-step method allows us to capture the curve's structure in tiny, manageable pieces. When these pieces are integrated through calculus, they provide the full arc length.

Integral Calculus

Integral calculus is a branch of mathematics that focuses on accumulation. It is essential in calculating areas, volumes, and even complex quantities like arc lengths. In this context, calculus serves as our tool to sum up many tiny bits to achieve a total.
In our arc length problem, we started by representing the arc as a series of short lines, and the sum of these lines gives an approximation of the curve's length. However, to move from approximation to precision, we let \(\delta x\) approach zero, resulting in a limit that forms an integral.
The step where we transition from a sum of short lines to a continuous integral is the heart of integral calculus. The integral expression for the arc length, \(\int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} dx\), combines the principles of derivatives (for calculating \(\frac{dy}{dx}\)) and integration.

• It calculates the smooth arc length across \[x=a\] to \[x=b\].
• Peeks into every teeny-tiny step along the curve and sums them up.

This method ensures that we strive for exactness, guided by the beauty of calculus.