All Flashcards

Area of a sector in polar coordinates?

$A = \frac{1}{2}r^2\theta$

Area enclosed by polar curve $r = f(\theta)$ from $a$ to $b$ ?

$A = \frac{1}{2}\int_{a}^{b}[f(\theta)]^2 d\theta$

Half-angle identity for $\cos^2(\theta)$ ?

$\cos^2(\theta) = \frac{1 + \cos(2\theta)}{2}$

Area of one petal of a rose curve?

$A = \frac{1}{2}\int_{a}^{b} [f(\theta)]^2 d\theta$ , where the limits a and b define one petal.

Why use polar coordinates for certain curves?

Easier to describe curves like circles and spirals than Cartesian coordinates.

Explain the concept of integrating in polar coordinates to find area.

Summing infinitely small sectors (pizza slices) to find the total area enclosed by the curve.

How does symmetry simplify area calculations in polar coordinates?

Calculate the area of one symmetric portion and multiply to get the total area.

Why is the area formula $\frac{1}{2}\int_{a}^{b}r^2 d\theta$ ?

It sums the areas of infinitesimal sectors with radius $r$ and angle $d\theta$ .

What does the integral $\int_{a}^{b} r d\theta$ represent in polar coordinates?

It does not directly represent area; the correct area integral is $\frac{1}{2}\int_{a}^{b} r^2 d\theta$ .

Define polar coordinates.

A system using radius (r) and angle (θ) to locate points.

What is a polar curve?

A curve defined by an equation in polar coordinates, typically r = f(θ).

Define a sector in the context of polar coordinates.

A 'slice' of a circle defined by an angle θ and radius r.

What is a limacon?

A polar curve described by the equation $r = a + bsin( heta)$ or $r = a + bcos( heta)$ .

Define the area element in polar coordinates.

Infinitesimal area element used in integration, given by $\frac{1}{2}r^2 d\theta$ .