Welcome to polar coordinates in calculus! Unlike Cartesian coordinates that use x and y, polar coordinates locate points using distance r from the origin and angle theta from the positive x-axis. The conversion formulas connect these two systems: x equals r cosine theta, y equals r sine theta. Let's see how a point moves as we change these polar coordinates.
Now let's explore common polar curves. The circle has equation r equals a constant. The cardioid, shaped like a heart, has equation r equals a times one plus cosine theta. Rose curves use r equals a cosine n theta, creating petal patterns. Spirals follow r equals a theta, expanding outward. Each curve has unique properties like symmetry and periodicity that help us analyze them.
To find derivatives in polar coordinates, we need the slope formula for tangent lines. The derivative dy over dx equals a fraction where the numerator is dr d-theta sine theta plus r cosine theta, and the denominator is dr d-theta cosine theta minus r sine theta. Let's see this in action with a cardioid where r equals one plus cosine theta. The derivative dr d-theta is negative sine theta.
To find areas in polar coordinates, we use the formula A equals one half times the integral from alpha to beta of r squared d-theta. This comes from summing up infinitesimal sectors. For a circle with radius 2, the area is one half times the integral from 0 to 2 pi of 4 d-theta, which equals 4 pi. Watch as we sweep through the angle to build up the area.
To summarize what we've learned about polar coordinates in calculus: We use distance r and angle theta to locate points. Common curves include circles, cardioids, and rose patterns. Finding derivatives requires the polar slope formula. Areas are calculated using the integral of one half r squared d-theta. These fundamental tools open the door to advanced calculus applications in physics and engineering.