Question

Find the area of the surface formed by revolving the curve about the given line. $$ \begin{array}{llll} \underline{\text { Polar Equation }} & & \text { Interval } & & \text { Axis of Revolution } \\ r=a(1+\cos \theta) & & 0 \leq \theta \leq \pi & & \text { Polar axis } \end{array} $$

Short answer

This exercise's accurate, complete solution will likely require numerical methods or a software tool to evaluate the integral, and the final answer will depend on the value of a.

Step-by-step solution

Define the function and calculate the derivative

The function given in polar coordinates is \(f(\theta) = a (1 + \cos(\theta))\). The derivative of the function w.r.t. theta is obtained as: \(\frac{{df}}{{d\theta}} = - a \sin(\theta)\)

Substitute into the area of revolution formula

Substitute \(f(\theta)\) and \(\frac{{df}}{{d\theta}}\) in the formula of the surface area of revolution: \(A = 2\pi \int_{0}^{\pi} a (1+\cos(\theta)) \sqrt{1 + (\sin(\theta))^2} d\theta\)

Calculate the integral

The resulting integral can be difficult to solve symbolically and in most cases, numerical methods (like Riemann sums, trapezoidal rule, Simpson's rule, etc.) or some software/tool might need to be used to compute it.

Key concepts

Polar Equations

When working in the realm of polar coordinates, we describe points on a plane using the distance from the origin and an angle from the reference direction (usually the positive x-axis). Polar equations like the one in our exercise, \( r = a(1 + \cos(\theta)) \), provide a relationship between \( r \), the radius (distance from the origin), and \( \theta \), the angle.

These equations can often yield intricate and beautiful curves - such as circles, spirals, and roses - that would be complex to describe using traditional Cartesian coordinates (x, y).

Understanding polar equations is crucial, especially when dealing with problems in physics or engineering where symmetry relative to a central point is present. To fully interpret these equations, one must consider both \( r \) and \( \theta \) as they define the position of a point in the polar plane.

Definite Integrals

Definite integrals are a cornerstone of calculus, serving as a tool to calculate the accumulation of quantities such as areas, volumes, and total change. They represent the net result of a continuous function over an interval, symbolized by integrals with limits.

For example, the integral from the exercise, \(2\pi \int_{0}^{\pi} a (1+\cos(\theta)) \sqrt{1 + (\sin(\theta))^2} d\theta\), asks for the surface area formed by revolving a curve over a specific range. The resulting number provides us with the exact measurement of that area.

Definite integrals give us the freedom to solve real-world problems that require precise computations such as in engineering, physics, and many other fields that rely on practical applications.

Calculus with Polar Coordinates

Calculus with polar coordinates allows us to tackle problems involving curves and areas that revolve around points, rather than being dependent on the straight-lined Cartesian system. This approach is a boon when the geometries involved are naturally circular or radial.

To solve problems like the exercise given, one must be comfortable converting between Cartesian and polar coordinates and applying the fundamental theorems of calculus in the polar context. For instance, the area of revolution requires knowing how to derive the polar function to apply it to the appropriate formula. It is also essential to integrate within the prescribed limits to calculate properties like the surface area or the arc length of curves described by polar equations.

Taking on calculus problems in polar coordinates often introduces additional challenges, such as more complicated integrals, but it provides a powerful method for certain symmetrical problems that are exceedingly tough to handle otherwise.