Question

Finding arc lengthFind the length of the curvey = ∫ from 0 to x of √(cos(2t)) dt, 0 ≤ x ≤ π/4.

Accepted Answer

Recognize that the curve is defined by an integral function: $$y = \$$int_0$$^x \sqrt{\cos(2t)} \, dt. To $$find the arc length of $$y$$ from $$x=0 to$$ $$x=\frac{\pi}{4}$$, we use the arc length formula for a function $$y=f(x)$$: $$L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\$$right)^2$$} \, dx. $$Find the derivative $$\frac{dy}{dx}$$ using the Fundamental Theorem of Calculus. Since $$y is $$defined as an integral with variable upper limit $$x$$, we have $$\frac{dy}{dx} = \sqrt{\cos(2x)}. $$Substitute $$\frac{dy}{dx}$$ into the arc length formula: $$L = \$$int_0$$^{\frac{\pi}{4}} \sqrt{1 + \left(\sqrt{\cos(2x)}\$$right)^2$$} \, dx. $$Simplify the expression inside the square root: $$\left(\sqrt{\cos(2x)}\$$right)^2$$ = \cos(2x)$$, so the integrand becomes $$\sqrt{1 + \cos(2x)}. $$Use a trigonometric identity to simplify $$1 + \cos(2x). $$Recall that $$1 + \cos(2x) = 2 \cos^2$(x). $$Therefore, the integrand simplifies to $$\sqrt{2 \cos^2$(x)} = \sqrt{2} |\cos(x)|. $$Since $$x is in$$ $$[0, \frac{\pi}{4}]$$ where $$\cos(x) is $$positive, the absolute value can be removed. The arc length integral becomes $$L = \$$int_0$$^{\frac{\pi}{4}} \sqrt{2} \cos(x) \, dx.$$

Finding arc length
Find the length of the curve
y = ∫ from 0 to x of √(cos(2t)) dt, 0 ≤ x ≤ π/4.

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Key Concepts

Arc Length Formula for Parametric and Integral-Defined Curves

Fundamental Theorem of Calculus

Handling Square Roots of Trigonometric Functions