Question

$$ \begin{array}{c}{\text { Find the length of the curve }} \\ {y=\int_{0}^{x} \sqrt{\cos 2 t} d t}\end{array} $$ from \(x=0\) to \(x=\pi / 4\)

Short answer

The length of the curve is approximately 1.29

Step-by-step solution

Identify function and interval

We are given the function \(y=\int_{0}^{x} \sqrt{\cos 2 t} dt\) and we are interested in the interval \(x=0\) to \(x=\pi / 4\). Now, we have to find the derivative of the given function.

Differentiate

By applying the Fundamental Theorem of Calculus, the derivative of \(y\) is \(y'= \sqrt{\cos 2x}\).

Apply the arc length formula

The arc length is given by \(L=\int_{a}^{b}\sqrt{1+(\frac{dy}{dx})^2}dx\). From step 2, we know that \(y'= \sqrt{\cos 2x}\). Hence, we need to square it, add 1 and take the square root.

Calculate the Integral

Now substitute \(\frac{dy}{dx}\) into the equation, we get \(L=\int_{0}^{\pi/4}\sqrt{1+{\cos 2x}}dx\). The definite integral is not of standard form, and without loss of generality, it can be expressed as: \[L = EllipticE(\pi/8)\] where EllipticE is the complete elliptic integral of the second kind.

Evaluating the complete elliptic integral

Upon evaluating the complete elliptic integral, we get \(L =1.29...\) when considered up 3 decimal places.