Question

$$ \begin{array}{l}{\text { True or False If a function } y=f(x) \text { is continuous on an }} \\ {\text { interval }[a, b], \text { then the length of its curve is given by }} \\ {\int_{a}^{b} \sqrt{1+\left(\frac{d y}{d x}\right)^{2}} d x . \text { Justify your answer. }}\end{array} $$

Short answer

The statement is true, given that the function \(y=f(x)\) is differentiable on the interval \([a,b]\).

Step-by-step solution

Understand the concept

The problem involves the concept of arc length of a curve. The given formula is that of the arc length, a fundamental concept in calculus. The general formula for arc length for a function \(y=f(x)\) that is differentiable on an interval \([a,b]\) is \(\int_{a}^{b} \sqrt{1+\left(\frac{d y}{d x}\right)^{2}} d x \). This formula comes from the Pythagorean theorem, applied to the infinitesimally small triangles formed by the function's curve.

Apply the concept to the given function

In this case, for a function \(y=f(x)\) that is continuous (and differentiable, as required by the formula) on an interval \([a,b]\), we can indeed calculate the length of the function's curve (or 'arc length') using the given formula. Therefore, the statement, as long as \(f(x)\) is differentiable, is true.

Provide full justification

To justify this answer, we must recall why the arc length formula works. It measures the 'length' of the function throughout the interval by summing up infinitesimal 'straight-line' distances, \(\sqrt{1+(\frac{d y}{d x})^{2}}dx\), between very close points on the function. A measurable length defined in this way requires the function to be not just continuous, but differentiable - without a well-defined gradient at every point in the interval, we cannot construct our infinitesimal triangles to measure the length. The function \(y=f(x)\) needs to be smooth (have a derivative) on the interval \([a,b]\) for the statement to be true.

Key concepts

Integral Calculus

Integral calculus is a branch of mathematics focusing on the accumulation of quantities and the areas under and between curves. When you're dealing with the arc length of a curve, you're essentially summing up an infinite number of infinitesimally small segments of a curve to find the total length. This is done using an integral, which is a mathematical tool that allows for the addition of an infinite number of tiny factors, segments, or areas to obtain a finite result.

To calculate the arc length of a curve represented by a function, you integrate over the interval where the function is defined. The arc length formula \(\int_{a}^{b} \sqrt{1+\left(\frac{d y}{d x}\right)^{2}} d x\) is a direct application of integral calculus. The integrand \(\sqrt{1+\left(\frac{d y}{d x}\right)^{2}}\) represents the length of the hypotenuse of an infinitesimally tiny right triangle formed when considering the curve's slope at a given point.

Differentiability

Differentiability refers to the smoothness of a function, specifically, whether the function has a derivative at every point within a given interval. A function is said to be differentiable if it has no sharp corners or discontinuities, and we can calculate its slope at any point. This is crucial for calculating arc length because the formula hinges on the idea that you can take the derivative of the function \(y=f(x)\) and square it.

When we say a curve is differentiable on \( [a, b]\), we mean we can find the derivative \(\frac{dy}{dx}\) at every point between 'a' and 'b', including the endpoints. For arc length, without a continuous derivative, our infinitesimally small triangles used in the Pythagorean theorem would not be well-defined, and so the arc length would be impossible to calculate correctly.

Function Continuity

Function continuity is concerned with whether or not a function has any interruptions or gaps in its domain. If a function is continuous on the interval \( [a, b]\), it means that the function is unbroken and there are no jumps, holes, or vertical asymptotes within that range. Continuity is a prerequisite for differentiability; a function must be continuous to be differentiable, but being continuous doesn't ensure differentiability.

A continuous function, however, can still have points where it isn't smooth, such as a 'cusps' or 'corners'. While these points break differentiability, they don't violate continuity. To safely use the arc length formula, continuity is not enough, yet it is one of the necessary steps toward establishing the smoothness required for differentiability.

Pythagorean Theorem in Calculus

The Pythagorean theorem in calculus is an extension of the traditional theorem you're likely familiar with from geometry. It states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This theorem helps in finding the lengths of curves by treating infinitesimally small segments of the curve as the hypotenuse of a right triangle.

In the arc length formula, \(\sqrt{1+\left(\frac{d y}{d x}\right)^2}\) serves as the length of the hypotenuse of these infinitely small triangles. The term \(1+\left(\frac{d y}{d x}\right)^2\) is the sum of the squares of the 'vertical' (dy or \(\frac{dy}{dx}\)) and 'horizontal' (dx) sides. The square root brings us back to the actual length of the hypotenuse, and we integrate this length from 'a' to 'b' to accumulate the total arc length of the curve.