PLEASE HELP! Which one or ones of the following integrals produces the area that lies inside the circle r = 3 sin θ and outside the cardioid r = 1 + sin θ? (10 points)

Question

PLEASE HELP! Which one or ones of the following integrals produces the area that lies inside the circle r = 3 sin θ and outside the cardioid r = 1 + sin θ? (10 points)

I. one half times the integral from pi over 6 to 5 times pi over 6 of the quantity 9 times the sine squared of theta minus the square of 1 plus sine theta, d theta
II. the integral from pi over 6 to pi over 2 of the quantity 9 times the sine squared of theta minus the square of 1 plus sine theta, d theta
III. one half times the integral from 0 to pi over 6 of the quantity 9 times the sine squared of theta minus the square of 1 plus sine theta, d theta plus one half times the integral from 5 times pi over 6 to pi of the quantity 9 times the sine squared of theta minus the square of 1 plus sine theta, d theta

A) I only
B) I and II only
C) II only
D) III only

Answer 1

You are correct Aria.

Answer 2

WHICH ONE IS IT. I am a little confused.

Answer 3

If you have

R = 3 sinθ
r = 1 + sin θ
then you want A = ∫1/2 (R^2-r^2) dθ

Sorry, I just can't see wading through all those words to see which matches.

Answer 4

To find the area that lies inside the circle r = 3 sin θ and outside the cardioid r = 1 + sin θ, we can use the concept of double integrals in polar coordinates.

To solve this problem, we need to determine the correct integral(s) that will give us the desired area. Let's analyze each option:

I. This option involves calculating one half times the integral from pi/6 to 5π/6 of the quantity 9 times the sine squared of theta minus the square of 1 plus sine theta, d theta.

II. This option involves calculating the integral from pi/6 to π/2 of the quantity 9 times the sine squared of theta minus the square of 1 plus sine theta, d theta.

III. This option involves calculating one half times the integral from 0 to pi/6 of the quantity 9 times the sine squared of theta minus the square of 1 plus sine theta, d theta, plus one half times the integral from 5π/6 to π of the quantity 9 times the sine squared of theta minus the square of 1 plus sine theta, d theta.

By analyzing the equations for the circle and the cardioid, we can determine that the desired area lies between θ = 0 and θ = π/2. This means that options I and III cannot be correct because they have integrals that extend beyond θ = π/2.

So, the correct option is C) II only, which is the integral from pi/6 to π/2 of the given quantity.

Therefore, the integral given in option II will produce the area that lies inside the circle r = 3 sin θ and outside the cardioid r = 1 + sin θ.

PLEASE HELP! Which one or ones of the following integrals produces the area that lies inside the circle r = 3 sin θ and outside the cardioid r = 1 + sin θ? (10 points)

yes

You are correct Aria.

WHICH ONE IS IT. I am a little confused.

If you have

To find the area that lies inside the circle r = 3 sin θ and outside the cardioid r = 1 + sin θ, we can use the concept of double integrals in polar coordinates.