# Find the area of a single loop in curve #r=\sin(6\theta)#?

##
I am told the formula is #A=1/2\int_a^br^2d\theta# , so

#A=1/2\int_a^b(\sin(6\theta))^2d\theta#

But what are the bound values, #a# and #b# ?

I am told the formula is

But what are the bound values,

The area of 1 loop of the given polar curve is

Start by drawing the polar curve. It helps to picture it.

As you can see, each loop starts and ends when

#sin(6theta) = 0#

#6theta = 0 or 6theta = pi#

#theta = 0 or theta = pi/6#

Thus we will be finding the value of

#A = 1/2int_0^(pi/6) sin^2(6x)dx#

Recall that

#\color(maroon)(A=1/2\int_0^(\pi/6)(1/2-\cos(12x)/2)dx)#

#\color(maroon)(A=1/2{:[1/2x-1/2(1/12\sin(12x))]|:}_0^(\pi/6))#

#A = 1/2{:[1/2x - 1/24sin(12x)]|:}_0^(pi/6)#

#A = 1/2(pi/12)#

#A = pi/24#

Hopefully this helps!

By signing up, you agree to our Terms of Service and Privacy Policy

To find the area of a single loop in the curve ( r = \sin(6\theta) ), integrate ( \frac{1}{2} r^2 ) from ( \theta = 0 ) to the point where ( r = 0 ) for the first time in one period. This gives the area of half a loop. Then, multiply the result by 2 to get the total area of one loop.

[ A = 2 \int_0^{\frac{\pi}{6}} \frac{1}{2} (\sin(6\theta))^2 , d\theta ]

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7