Area of Polar Curves: Find & Calculate with Step-by-Step Guide

[knv]

1. Given the curves r = 2sin(θ) and r = 2sin(2θ), 0≤θ≤π/2, find the area of the region outside the first curve and inside the second curve




2.not sure which equations to use



3. I got 1 and 1/2 as the area and they were wrong. I do not really know how to work this problem. A detailed step by step explanation would be really helpful!

 

[tiny-tim]

hi knv! welcome to pf!

show us your full calculations, and then we'll see what went wrong, and we'll know how to help!

 

[knv]

Well I don't exactly know how to even start this problem but let's see.


∫2sin(2θ) - ∫2 sin(θ)

do the bounds change for the first one to π/4? Any hints on how to even start this problem. I am not looking for an answer. I do not understand my books explanation so I am really lost

 

[LCKurtz]

Well I don't exactly know how to even start this problem but let's see.


∫2sin(2θ) - ∫2 sin(θ)

do the bounds change for the first one to π/4? Any hints on how to even start this problem. I am not looking for an answer. I do not understand my books explanation so I am really lost

Generally the area inside a polar graph is $\int_a^b \frac 1 2 r^2\, d\theta$ between appropriate limits $a$ and $b$. If you want the area between two graphs the integrand would be $\frac 1 2( r_{outer}^2-r_{inner}^2)$. So you should sketch the two curves in the first quadrant and determine where they intersect and what the requestion region looks like. Then use the formula.

 

[knv]

∫sin²(θ) - ∫ sin²(2θ)

since sin² = 1/2(1-cos 2θ) we plug it in and bring the 1/2 to the front of each integral giving us..

1/2∫(1-cos 2θ) - 1/2∫(1-cos 4θ)

am I heading in the right direction? how do I find the bounds?

 

[LCKurtz]

∫sin²(θ) - ∫ sin²(2θ)

since sin² = 1/2(1-cos 2θ) we plug it in and bring the 1/2 to the front of each integral giving us..

1/2∫(1-cos 2θ) - 1/2∫(1-cos 4θ)

am I heading in the right direction? how do I find the bounds?

You follow my suggestion and plot the two curves in the first quadrant. How else will you know which is the inner and outer curve and what the area looks like? To find where they intersect set the $r$'s equal and find out which $\theta$ in the first quadrant does that.

 

[knv]

Im still having trouble with this problem. Can anyone help? Please I am so lost. Just need some help on setting it up and getting it started

 

[Mark44]

You've been given good advice about sketching the graphs of the two functions. Have you done that yet?

 

[knv]

Yeah I did that. And the inner one is 2 sin θ and the outer one is 2 sin 2θ

They cross at 0 and π/4

Correct?

 

[knv]

can I solve for half of the area using only one of the functions and then doubling it?

 

[knv]

I got the answer 1/4. Does anyone know if that is correct before I use my last attempt?

 

[LCKurtz]

Yeah I did that. And the inner one is 2 sin θ and the outer one is 2 sin 2θ

They cross at 0 and π/4

Correct?

can I solve for half of the area using only one of the functions and then doubling it?

No, why would you think that unless you are just guessing?

If you know the outer curve and inner curve and where they cross, you know everything you need to know to use the formula I gave you in post #4. Try it.

 

[knv]

No i had watched a video where a guy did that so I wasn't sure if it would work in this case.

I worked it out doing what you told me to do and came up with the answer 1/4.

Can you tell me if that is correct?


∫0->π/4 = ∫1/2 (4sin² 2θ) dθ -∫1/2(2 sin ²θ) dθ

2∫(1/2)(1-cos4θ)dθ - ∫(1/2)(1-cos 2θ) dθ

[(θ-sin4θ)/4] - 1/2[(θ-sin 2θ)/2]

(π/4-0)/4) - 1/2[(π/4 - 4/4)/2]

so it would end up being

π/16 - π/16 +1/4

 

[LCKurtz]

You are close. The answer should be 1/2, so you are off by a factor of 2.
Are you sure you got everything squared that is supposed to be in your setup?

 

[knv]

I had found 1/2 earlier and it was counted wrong.

 

[LCKurtz]

I had found 1/2 earlier and it was counted wrong.

I didn't notice you had the intersection point wrong. Please show me the steps how you got $\pi/4$.

 

[SammyS]

Yeah I did that. And the inner one is 2 sin θ and the outer one is 2 sin 2θ

They cross at 0 and π/4

Correct?

No.

They intersect at θ = 0 and θ = π/3 .