63–74. Arc length of polar curves Find the length of the following polar curves.

The complete circle r = a sin θ, where a > 0

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Recall the formula for the arc length \( L \) of a curve given in polar coordinates \( r = r(\theta) \) from \( \theta = \alpha \) to \( \theta = \beta \): \[ L = \int_{\alpha}^{\beta} \sqrt{r(\theta)^2 + \left(\frac{d r}{d \theta}\right)^2} \, d\theta \]

Identify the given polar curve: \( r = a \sin \theta \), where \( a > 0 \). Since this represents a complete circle, the parameter \( \theta \) will vary from \( 0 \) to \( \pi \) to trace the entire circle.

Compute the derivative of \( r \) with respect to \( \theta \): \[ \frac{d r}{d \theta} = a \cos \theta \]

Substitute \( r \) and \( \frac{d r}{d \theta} \) into the arc length formula: \[ L = \int_0^{\pi} \sqrt{(a \sin \theta)^2 + (a \cos \theta)^2} \, d\theta \] Simplify the expression inside the square root before integrating.

Evaluate the integral over the interval \( 0 \leq \theta \leq \pi \) to find the total length of the curve, which corresponds to the circumference of the circle described by \( r = a \sin \theta \).

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주요 개념

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Polar Coordinates and Polar Curves

Polar coordinates represent points in the plane using a radius and an angle (r, θ). Polar curves are defined by equations relating r and θ, such as r = a sin θ, which describes a circle. Understanding how to interpret and plot these curves is essential for analyzing their properties.

Arc Length Formula for Polar Curves

The arc length of a polar curve r(θ) from θ = α to θ = β is given by the integral ∫ from α to β of √[r(θ)² + (dr/dθ)²] dθ. This formula accounts for changes in both radius and angle, allowing calculation of the curve's length in polar form.

Differentiation of Polar Functions

To apply the arc length formula, one must compute the derivative dr/dθ of the polar function r(θ). This involves differentiating trigonometric functions like sin θ, which is crucial for evaluating the integral and finding the exact length of the curve.

63–74. Arc length of polar curves Find the length of the following polar curves.The complete circle r = a sin θ, where a > 0