Q: Which formula gives the area $$A of a $$region bounded by a polar curve $$r = f(\theta)$$ from $$\theta = a to$$ $$\theta = b$$?

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

Q: Suppose you want to find the area inside the circle $$r = 3\sin\theta$$ and outside the cardioid $$r = 1 + \sin\theta. $$Which integral correctly represents this area?

$$A = \frac{1}{2} \$$int_0$$^{\pi} \left( [3\sin\theta]^2 - [1 + \sin\theta]^2 \right) d\theta$$

Q: What is the formula for the arc length $$L of a $$curve given by $$r = f(\theta)$$ from $$\theta = a to$$ $$\theta = b$$?

$$L = \int_a^b \sqrt{[f(\theta)]^2 + \left( \frac{dr}{d\theta} \$$right)^2$$} d\theta$$

Q: Given the polar curve $$r = 2 + 2\cos\theta$$, what is the correct integral to find its arc length from $$\theta = -\pi to$$ $$\theta = \pi$$?

$$L = \int_{-\pi}^{\pi} \sqrt{[2 + 2\cos\theta]^2 + [-2\sin\theta]^2} d\theta$$

Q: Which of the following best describes the process for finding the area between two polar curves $$r = f(\theta)$$ and $$r = g(\theta)$$ from $$\theta = a to$$ $$\theta = b$$?

$$A = \frac{1}{2} \int_a^b \left( [f(\theta)]^2 - [g(\theta)]^2 \right) d\theta$$

Q: A biologist models the boundary of a pond by $$r = 2 + \sin\theta$$ for $$0 \leq \theta \leq 2\pi. $$What is the formula for the area enclosed by the pond?

$$A = \frac{1}{2} \$$int_0$$^{2\pi} [2 + \sin\theta]^2 d\theta$$

Q: For the curve $$r = \cos 2\theta$$, what is the area enclosed by one loop?

$$A = \frac{1}{2} \$$int_0$$^{\pi/2} [\cos 2\theta]^2 d\theta$$

Q: A physicist wants to find the area swept out by a rotating radar antenna described by $$r = 5\cos\theta$$ from $$\theta = 0 to$$ $$\theta = \frac{\pi}{2}. $$What is the correct integral for the area?

$$A = \frac{1}{2} \$$int_0$$^{\pi/2} [5\cos\theta]^2 d\theta$$

Q: Which of the following is the correct expression for the arc length of $$r = f(\theta)$$ from $$\theta = a to$$ $$\theta = b$$?

$$L = \int_a^b \sqrt{[f(\theta)]^2 + \left( \frac{df}{d\theta} \$$right)^2$$} d\theta$$

Q: A medical imaging device scans a region described by $$r = 4\sin\theta$$ for $$0 \leq \theta \leq \pi. $$What is the area scanned?

$$A = \frac{1}{2} \$$int_0$$^{\pi} [4\sin\theta]^2 d\theta$$

Question 1

Which formula gives the area $$A of a $$region bounded by a polar curve $$r = f(	heta)$$ from $$	heta = a to$$ $$	heta = b$$?

Accepted Answer

$$A = \frac{1}{2} \int_a^b [f(	heta)]^2 d	heta$$

Question 2

Suppose you want to find the area inside the circle $$r = 3\sin	heta$$ and outside the cardioid $$r = 1 + \sin	heta. $$Which integral correctly represents this area?

Accepted Answer

$$A = \frac{1}{2} \$$int_0$$^{\pi} \left( [3\sin	heta]^2 - [1 + \sin	heta]^2 ight) d	heta$$

Question 3

What is the formula for the arc length $$L of a $$curve given by $$r = f(	heta)$$ from $$	heta = a to$$ $$	heta = b$$?

Accepted Answer

$$L = \int_a^b \sqrt{[f(	heta)]^2 + \left( \frac{dr}{d	heta} \$$right)^2$$} d	heta$$

Question 4

Given the polar curve $$r = 2 + 2\cos	heta$$, what is the correct integral to find its arc length from $$	heta = -\pi to$$ $$	heta = \pi$$?

Accepted Answer

$$L = \int_{-\pi}^{\pi} \sqrt{[2 + 2\cos	heta]^2 + [-2\sin	heta]^2} d	heta$$

Question 5

If $$r = \cos 2	heta$$ and $$r = \frac{1}{2}$$, how many points of intersection do these curves have in $$0 \leq 	heta \u003c 2\pi$$?

Accepted Answer

4

Question 6

Which of the following best describes the process for finding the area between two polar curves $$r = f(	heta)$$ and $$r = g(	heta)$$ from $$	heta = a to$$ $$	heta = b$$?

Accepted Answer

$$A = \frac{1}{2} \int_a^b \left( [f(	heta)]^2 - [g(	heta)]^2 ight) d	heta$$

Question 7

A biologist models the boundary of a pond by $$r = 2 + \sin	heta$$ for $$0 \leq 	heta \leq 2\pi. $$What is the formula for the area enclosed by the pond?

Accepted Answer

$$A = \frac{1}{2} \$$int_0$$^{2\pi} [2 + \sin	heta]^2 d	heta$$

Question 8

For the curve $$r = \cos 2	heta$$, what is the area enclosed by one loop?

Accepted Answer

$$A = \frac{1}{2} \$$int_0$$^{\pi/2} [\cos 2	heta]^2 d	heta$$

Question 9

A physicist wants to find the area swept out by a rotating radar antenna described by $$r = 5\cos	heta$$ from $$	heta = 0 to$$ $$	heta = \frac{\pi}{2}. $$What is the correct integral for the area?

Accepted Answer

$$A = \frac{1}{2} \$$int_0$$^{\pi/2} [5\cos	heta]^2 d	heta$$

Question 10

Which of the following is the correct expression for the arc length of $$r = f(	heta)$$ from $$	heta = a to$$ $$	heta = b$$?

Accepted Answer

$$L = \int_a^b \sqrt{[f(	heta)]^2 + \left( \frac{df}{d	heta} \$$right)^2$$} d	heta$$

Question 11

A medical imaging device scans a region described by $$r = 4\sin	heta$$ for $$0 \leq 	heta \leq \pi. $$What is the area scanned?

Accepted Answer

$$A = \frac{1}{2} \$$int_0$$^{\pi} [4\sin	heta]^2 d	heta$$

Question 12

If $$r = f(	heta) is a $$positive continuous function on $$[a, b]$$, which of the following statements is true about the area enclosed by the curve?

Accepted Answer

The area is always $$\frac{1}{2} \int_a^b [f(	heta)]^2 d	heta$$

Question 13

A chemist is analyzing a reaction region bounded by $$r = 2$$ and $$r = 2\sin	heta$$ for $$0 \leq 	heta \leq \pi. $$What is the correct integral for the area inside $$r = 2$$ and outside $$r = 2\sin	heta$$?

Accepted Answer

$$A = \frac{1}{2} \$$int_0$$^{\pi} [2^2$ - (2\sin\$$theta)^2$$] d	heta$$

Question 14

Which of the following is a correct step in approximating the area of a region bounded by $$r = f(	heta)$$ using sectors?

Accepted Answer

Divide the interval $$[a, b]$$ into $$n$$ equal subintervals and approximate each sector's area by $$\frac{1}{2} [f(	heta_i)]^2 \Delta 	heta$$

Question 15

A meteorologist models a storm region by $$r = 3 + 2\cos	heta$$ for $$0 \leq 	heta \leq 2\pi. $$What is the correct formula for the area covered by the storm?

Accepted Answer

$$A = \frac{1}{2} \$$int_0$$^{2\pi} [3 + 2\cos	heta]^2 d	heta$$

Question 16

Which of the following is the correct formula for the area of a sector of a circle with radius $$r$$ and central angle $$\Delta 	heta (in $$radians)?

Accepted Answer

$$A = \frac{1}{2} r^2$ \Delta 	heta$$

Question 17

A botanist is studying a flower with a boundary given by $$r = \cos 3	heta. $$What is the area enclosed by one petal?

Accepted Answer

$$A = \frac{1}{2} \$$int_0$$^{\pi/3} [\cos 3	heta]^2 d	heta$$

Question 18

Which of the following is true about the area inside $$r = f(	heta)$$ and outside $$r = g(	heta)$$ for $$	heta in$$ $$[a, b]$$?

Accepted Answer

$$A = \frac{1}{2} \int_a^b ([f(	heta)]^2 - [g(	heta)]^2) d	heta$$

Section 1.4 - Calculus in Polar Coordinates

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