How do you find the area of one petal of #r=2cos3theta#?
First, graph
Next, using either a graphing utility or this graph paper, plot the graph using convenient points.
The area of a petal can be determined by an integral of the form
Notice the petal in Quadrant I and IV does not extend past
Letting the interval of integration go from
The halfangle identity for cosine says
This changes our integral to
Using
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To find the area of one petal of the polar curve ( r = 2 \cos(3\theta) ), follow these steps:

Determine the limits of integration for ( \theta ) that correspond to one complete petal. This typically involves finding the values of ( \theta ) where the curve intersects itself.

Set up the integral for the area using the formula for the area enclosed by a polar curve:
[ A = \frac{1}{2} \int_{\theta_1}^{\theta_2} [r(\theta)]^2 , d\theta ]
where ( r(\theta) ) is the equation of the curve and ( \theta_1 ) and ( \theta_2 ) are the limits of integration.

Substitute ( r(\theta) = 2 \cos(3\theta) ) into the integral.

Integrate the expression obtained in step 3 with respect to ( \theta ) from ( \theta_1 ) to ( \theta_2 ).

Evaluate the integral to find the area of one petal of the curve.
By following these steps, you can find the area of one petal of the polar curve ( r = 2 \cos(3\theta) ).
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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