What is the area enclosed between the two polar curves: #r = 4 - 2cos 3theta# and #r = 5# ?
for vivid clear illustration in his answer) can be regarded as the
cos^3A=1/4(cos 9A+3 cos 3A)#
Altogether in the four quadrants, the area is
I think this shaded area is referred to as in between area. Here,
the circle is the exterior curve.
There is a set of three other equal in-between areas, for which
the circle is the interior. curve.
make it for Sam....
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#"area" = (7pi)/9 +(17sqrt(3))/12#
There are a couple of ingredients to this:
Given two curves:
Points of intersection will satisfy:
Hence:
The area is similar to that of a triangle with vertices:
which will be:
Whereas:
OK (finally)
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To find the area enclosed between the two polar curves ( r = 4 - 2\cos(3\theta) ) and ( r = 5 ), you need to set up the integral and evaluate it.
First, find the points of intersection of the two curves by equating them:
[ 4 - 2\cos(3\theta) = 5 ]
Solve for ( \theta ) to find the limits of integration.
Once you have the limits of integration, set up the integral for the area enclosed between the two curves:
[ A = \frac{1}{2} \int_{\theta_1}^{\theta_2} [r_2^2 - r_1^2] d\theta ]
where ( r_2 ) is the outer curve and ( r_1 ) is the inner curve.
Substitute the equations of the curves and integrate within the determined limits to find the area.
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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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