Question

Find the area inside the cardioid defined by the equation \(r=1-\cos \theta\).

Short answer

The area is \(\frac{3\pi}{2}\).

Step-by-step solution

Understand the Problem

A cardioid, in polar coordinates, is defined by the equation \(r = 1 - \cos \theta\). We need to find the area enclosed by this cardioid.

Use the Polar Area Formula

The formula for the area \(A\) enclosed by a curve defined by \(r = f(\theta)\) from \(\theta = a\) to \(\theta = b\) is \(A = \frac{1}{2} \int_{a}^{b} r^2 d\theta\). Here, \(r = 1 - \cos \theta\) and \(\theta\) ranges from \(0\) to \(2\pi\).

Set Up the Integral

Substitute \(r = 1 - \cos \theta\) into the formula: \[A = \frac{1}{2} \int_{0}^{2\pi} (1 - \cos \theta)^2 d\theta\].

Expand the Expression

Expand \((1 - \cos \theta)^2\): \[(1 - \cos \theta)^2 = 1 - 2\cos \theta + \cos^2 \theta\].

Apply Trigonometric Identities

Use the identity \(\cos^2 \theta = \frac{1 + \cos 2\theta}{2}\), so the expression becomes: \[1 - 2\cos \theta + \frac{1 + \cos 2\theta}{2}\]. Simplify this to \[\frac{3}{2} - 2\cos \theta + \frac{1}{2}\cos 2\theta\].

Evaluate the Integral

The integral becomes: \[A = \frac{1}{2} \int_{0}^{2\pi} \left(\frac{3}{2} - 2\cos \theta + \frac{1}{2}\cos 2\theta\right) d\theta\]. Separate this into: \[\frac{1}{2}\left( \int_{0}^{2\pi} \frac{3}{2} d\theta - 2 \int_{0}^{2\pi} \cos \theta d\theta + \frac{1}{2} \int_{0}^{2\pi} \cos 2\theta d\theta \right)\].

Solve the Integral Components

Evaluate each integral: 1. \( \int_{0}^{2\pi} \frac{3}{2} d\theta = \frac{3}{2} \times 2\pi = 3\pi \)2. \( \int_{0}^{2\pi} \cos \theta d\theta = 0 \) (since \(\cos \theta\) is periodic and symmetric around zero)3. \( \int_{0}^{2\pi} \cos 2\theta d\theta = 0 \) (similar reasoning as above).

Finalize the Calculation

Substitute back into the area equation: \[A = \frac{1}{2} (3\pi - 0 + 0) = \frac{3\pi}{2}\].

Conclude the Solution

The area inside the cardioid defined by \(r = 1 - \cos \theta\) is \(\frac{3\pi}{2}\).

Key concepts

Polar Coordinates

Polar coordinates offer a way to describe the location of a point in the plane using the angle and distance from a fixed point called the pole. This is different from Cartesian coordinates, which use horizontal and vertical distances. In the case of polar coordinates, points are typically represented by \(r, \theta\), where \(r\) is the distance from the pole, and \(\theta\) is the angle from the positive x-axis.

Polar coordinates are particularly useful when dealing with curves like circles and spirals, where using angles and radii simplifies calculations. For example, the cardioid we are examining is described by the equation \(r = 1 - \cos \theta\). This showcases how the curve's shape hinges on changes in \(\theta\), marking polar coordinates as an ideal choice.

The entire plot of the cardioid occurs as \(\theta\) ranges from \([0, 2\pi]\), creating a loop that reflects the heart-like shape typical of a cardioid. Each point on this curve can be pinpointed using its corresponding \(\theta\) and \(r\), emphasizing the elegance and simplicity polar coordinates bring to problems involving rotations and curves.

Integral Calculus

Integral calculus enables the calculation of areas, volumes, and other quantities by breaking down a space into infinitely small parts. In this context, we use integral calculus to find the area enclosed by the cardioid defined in polar coordinates.

To determine the area of a region bounded by a polar curve \(r = f(\theta)\) over an interval of angles from \(a\) to \(b\), we employ the polar area formula:

• \[A = \frac{1}{2} \int_{a}^{b} r^2 d\theta\]

This formula is derived from considering the area of small sectors, much like slices of a pie, summed over the interval \(\theta\).

In our cardioid problem, substituting \(r = 1 - \cos \theta\) into the formula provides us with the starting point for calculating the area. The process involves setting up an integral over \(\theta\) from 0 to \(2\pi\), leading to an expression with various trigonometric functions that need to be simplified and evaluated.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all related angles. They are vital in simplifying complex expressions in calculus.

For the cardioid problem, we rely on the identity for \(\cos^2 \theta\):

• \[\cos^2 \theta = \frac{1 + \cos 2\theta}{2}\]

This identity helps in transforming the integrand \( (1 - \cos \theta)^2 \) into a more manageable form. Applying this identity allows us to break down trigonometric functions into simpler components, easing the way for the evaluation of integrals.

Simplifying the function using these identities transforms the expression into: \[\frac{3}{2} - 2\cos \theta + \frac{1}{2}\cos 2\theta\].

Understanding and applying trigonometric identities simplify the calculations, making it easier to work through the integral calculus required to find the area of the cardioid.