Question

Find the length of the curve over the given interval. $$ \begin{array}{ll} \text { Polar Equation } & \text { Interval } \\ r=2 a \cos \theta & -\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2} \\ \end{array} $$

Short answer

The length of the curve over the given interval is \( 2 \pi a \).

Step-by-step solution

Understand the given polar equation

The given polar equation \( r=2a \cos \theta \) describes a circle with radius \( 2a \) at origin in polar co-ordinates system where \( a \) is a constant. The range of \( \theta \) from \( -\pi/2 \) to \( \pi/2 \) determines that it covers the right half of the circle.

Derive the equation

To find the rate of change of \( r \) with respect to \( \theta \), differentiate \( r \) with respect to \( \theta \), which yields \( dr/d\theta = -2a \sin \theta \).

Use the formula for finding length of curve

Apply the formula for the length of a curve in polar notation, \( L = \int_{- \pi /2}^{\pi /2} \sqrt{(2a \cos \theta )^2 + (-2a \sin \theta )^2} d\theta \). Simplifying this, it becomes \( L = \int_{- \pi /2}^{\pi /2} \sqrt{(2a)^2} d\theta \) and finally, \( L = \int_{- \pi /2}^{\pi /2} 2a d\theta \).

Solve the integral

Solve the integral \( L = 2a \int_{- \pi /2}^{\pi /2} d\theta \). The solution of the integral yields the length of the curve as \( L = 2a (\pi /2 - (-\pi /2)) = 2a*\pi = 2 \pi a \).

Key concepts

Polar Equation

A polar equation is a way of expressing a two-dimensional curve using the polar coordinate system. Unlike the Cartesian coordinate system, which uses a grid of 'x' and 'y' values, polar coordinates are based on a distance from a central point (known as the radial coordinate 'r') and an angle from a reference direction (known as the angular coordinate '\theta'). The relationship between 'r' and '\theta' represents the shape of the curve.

For example, the polar equation r=2a \times \text{{cos}} \theta describes a circle if 'a' is a constant. This circle will have a radius of '2a', centered at the origin of the polar coordinate system. The interval (or domain) given for '\theta', in this case, from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \) tells us which portion of the circle is being considered; here, it is the right half of the circle. Understanding the polar equation is crucial to visualize the curve before proceeding to calculate its properties, such as length.

Differentiation in Polar Coordinates

Differentiation in polar coordinates is similar to differentiation in Cartesian coordinates, but it requires considering the rate of change of both the radius 'r' and the angle '\theta'. When working with polar equations, the derivative \(\frac{dr}{d\theta}\) is often needed, especially when calculating the length of a curve or other aspects that involve rates of change.

In our example, the derivative of the polar equation \(r=2a \text{{cos}} \theta\) with respect to '\theta' is found by applying standard differentiation rules to the cosine function. This yields \(\frac{dr}{d\theta} = -2a \text{{sin}} \theta\), which represents how rapidly the radius 'r' changes as '\theta' changes. This derivative is a key component in determining the length of a curve, as it will be part of the integrand in the length formula.

Integral Calculus

Integral calculus is a branch of mathematics focused on finding the area under curves, accumulation of quantities, and the lengths of curves, among other things. The length of a curve in polar coordinates is found using a specific integral formula that takes both 'r' and its derivative with respect to '\theta' into account.

The length 'L' of a polar curve from '\theta=a' to '\theta=b' is given by the integral \( L = \int_{a}^{b} \sqrt{(\frac{dr}{d\theta})^2 + (r)^2} d\theta \). In our exercise, after substituting and simplifying the integrand, the length is expressed as \( L = 2a \int_{-\pi /2}^{\pi /2} d\theta \), a straightforward integral with respect to '\theta'. Solving this integral gives a result proportional to the difference in the limits of integration, resulting in the total length of the curve as \( L = 2\pi a \), which represents the perimeter of the right half of the circle described by the polar equation.