Question

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

Short answer

True, the statement holds as the function provided meets all necessary conditions to apply the standard formula for arc length.

Step-by-step solution

Understanding the formula

The length of a curve of a function can be calculated by the arc length formula. The formula \(\int_{a}^{b} \sqrt{1+\left(\frac{d y}{d x}\right)^{2}} d x\) is the standard formula for finding the length of a curve from 'a' to 'b' for a function y = f(x). \(\frac{d y}{d x}\) is the derivative of the function.

Applying the correct conditions

This formula holds true under the condition that the function \(y=f(x)\) is not only differentiable, which implies it's smooth and has no sharp turns, but also continuous, which means there are no jumps or breaks in the curve. These conditions fall in line with the outlined state of function in problem - \(y=f(x)\) is differentiable on interval [a, b] - hence it should be continuous on [a, b] as well.

Drawing conclusion

Given that the function y=f(x) meets the relevant criteria to apply the formula mentioned, it can be stated that indeed the formula \(\int_{a}^{b} \sqrt{1+\left(\frac{d y}{d x}\right)^{2}} d x\) correctly calculates the length of its curve. Therefore, the statement is True.