How do you find the area of the region bounded by the polar curve #r=2+cos(2theta)# ?

Answer 1

The area inside a polar curve is approximately the sum of lots of skinny wedges that start at the origin and go out to the curve, as long as there are no self-intersections for your polar curve.

Each wedge or slice or sector is like a triangle with height #r# and base #r# #dθ#, so the area of each element is #dA = 1/2 b h = 1/2 r (r dθ) = 1/2 r^2 dθ.#

So add them up as an integral going around from θ=0 to θ=2π, and using a double angle formula, we get:

#A = 1/2 int_0 ^(2π)(2 + cos(2θ))^2 dθ#
#A = 1/2 int_0 ^(2π) [4 + 4 cos(2θ) + cos^2(2θ)] dθ#
#A = 1/2 int_0 ^(2π) [4 + 4 cos(2θ) + (1 + cos(4θ))/2] dθ.#

Now do the integral(s) by subbing u = 2θ and then u = 4θ, and remember to change limits for the "new u." I'll let you evaluate those to get practice integrating! Remember our motto,

"Struggling a bit makes you stronger." \dansmath/

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Answer 2

To find the area of the region bounded by the polar curve ( r = 2 + \cos(2\theta) ), you can use the formula for finding the area enclosed by a polar curve, which is:

[ A = \frac{1}{2} \int_{\alpha}^{\beta} [f(\theta)]^2 , d\theta ]

Where ( f(\theta) ) represents the polar curve equation ( r = 2 + \cos(2\theta) ), and ( \alpha ) and ( \beta ) are the angles where the curve starts and ends.

First, you need to find the values of ( \theta ) where the curve starts and ends, which is usually done by finding the points of intersection between the curve and itself (i.e., where ( r = 2 + \cos(2\theta) ) intersects itself).

Then, integrate ( [2 + \cos(2\theta)]^2 ) with respect to ( \theta ) from the starting angle to the ending angle, and multiply the result by ( \frac{1}{2} ).

This will give you the area of the region bounded by the polar curve.

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Answer from HIX Tutor

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.

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.

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