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  2. AP Calculus
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Steps to find area inside r=2cos⁡(θ)r = 2\cos(\theta)r=2cos(θ) from 000 to π\piπ?

  1. Set up: A=12∫0π(2cos⁡(θ))2dθA = \frac{1}{2}\int_{0}^{\pi}(2\cos(\theta))^2 d\thetaA=21​∫0π​(2cos(θ))2dθ. 2. Simplify. 3. Integrate and evaluate.
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Steps to find area inside r=2cos⁡(θ)r = 2\cos(\theta)r=2cos(θ) from 000 to π\piπ?

  1. Set up: A=12∫0π(2cos⁡(θ))2dθA = \frac{1}{2}\int_{0}^{\pi}(2\cos(\theta))^2 d\thetaA=21​∫0π​(2cos(θ))2dθ. 2. Simplify. 3. Integrate and evaluate.

Steps to find area of one petal of r=sin⁡(2θ)r = \sin(2\theta)r=sin(2θ)?

  1. Find range for one petal (e.g., 000 to π2\frac{\pi}{2}2π​). 2. Set up integral: A=12∫0π2(sin⁡(2θ))2dθA = \frac{1}{2}\int_{0}^{\frac{\pi}{2}}(\sin(2\theta))^2 d\thetaA=21​∫02π​​(sin(2θ))2dθ. 3. Integrate.

How to find the area of a polar curve when symmetry is present?

  1. Identify symmetry. 2. Find the limits of integration for one symmetrical part. 3. Integrate and multiply by the appropriate factor.

Steps to calculate area enclosed by r=3+3sin⁡(θ)r=3+3\sin(\theta)r=3+3sin(θ)?

  1. Sketch. 2. Recognize symmetry. 3. Set up integral: A=12∫02π(3+3sin⁡(θ))2dθA = \frac{1}{2}\int_{0}^{2\pi}(3+3\sin(\theta))^2 d\thetaA=21​∫02π​(3+3sin(θ))2dθ. 4. Simplify and solve.

How to deal with sin⁡2(θ)\sin^2(\theta)sin2(θ) or cos⁡2(θ)\cos^2(\theta)cos2(θ) in polar area integrals?

Use the half-angle identities: cos⁡2(θ)=1+cos⁡(2θ)2\cos^2(\theta) = \frac{1 + \cos(2\theta)}{2}cos2(θ)=21+cos(2θ)​ and sin⁡2(θ)=1−cos⁡(2θ)2\sin^2(\theta) = \frac{1 - \cos(2\theta)}{2}sin2(θ)=21−cos(2θ)​.

What is the first step in finding the area enclosed by a polar curve?

Sketch the curve to understand its shape and any symmetries.

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 12∫abr2dθ\frac{1}{2}\int_{a}^{b}r^2 d\theta21​∫ab​r2dθ?

It sums the areas of infinitesimal sectors with radius rrr and angle dθd\thetadθ.

What does the integral ∫abrdθ\int_{a}^{b} r d\theta∫ab​rdθ represent in polar coordinates?

It does not directly represent area; the correct area integral is 12∫abr2dθ\frac{1}{2}\int_{a}^{b} r^2 d\theta21​∫ab​r2dθ.

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)r = a + bsin( heta)r=a+bsin(heta) or r=a+bcos(heta)r = a + bcos( heta)r=a+bcos(heta).

Define the area element in polar coordinates.

Infinitesimal area element used in integration, given by 12r2dθ\frac{1}{2}r^2 d\theta21​r2dθ.