Questions: a. Set up an integral for the length of the curve. b. Graph the curve to see what it looks like. c. Use a grapher's or computer's integral evaluator to find the curve's length numerically. x=2 sin y, (π/6) ≤ y ≤ (5π/6) a. The integral is (Type an exact answer, using π and radicals as needed.)

Solution Steps

To find the length of the curve given by \( x = 2 \sin y \) for \( \frac{\pi}{6} \leq y \leq \frac{5\pi}{6} \), we use the formula for the arc length of a function \( x = f(y) \):

\[ L = \int_{a}^{b} \sqrt{1 + \left( \frac{dx}{dy} \right)^2} \, dy \]

First, we find \( \frac{dx}{dy} \):

\[ \frac{dx}{dy} = 2 \cos y \]

Then, the integrand becomes:

\[ \sqrt{1 + \left( 2 \cos y \right)^2} = \sqrt{1 + 4 \cos^2 y} \]

Thus, the integral for the length of the curve is:

\[ L = \int_{\frac{\pi}{6}}^{\frac{5\pi}{6}} \sqrt{1 + 4 \cos^2 y} \, dy \]

Final Answer