What is the area enclosed by #r=3sin(2theta(2pi)/4) theta# between #theta in [pi/8,(3pi)/4]#?
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To find the area enclosed by the polar curve ( r = 3\sin(2\theta  \frac{2\pi}{4})  \theta ) between ( \theta = \frac{\pi}{8} ) and ( \theta = \frac{3\pi}{4} ), you need to integrate ( \frac{1}{2}r^2 d\theta ) over the given interval.
[ A = \frac{1}{2} \int_{\frac{\pi}{8}}^{\frac{3\pi}{4}} (3\sin(2\theta  \frac{2\pi}{4})  \theta)^2 d\theta ]
Calculate this integral to find the area.
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To find the area enclosed by the polar curve ( r = 3\sin(2\theta  \frac{2\pi}{4})  \theta ) between ( \theta ) in ( [\frac{\pi}{8}, \frac{3\pi}{4}] ):

Identify the intersection points of the curve with itself by setting the equation equal to itself: [ 3\sin(2\theta  \frac{2\pi}{4})  \theta = 3\sin(2\theta  \frac{2\pi}{4})  \theta ] [ 0 = 0 ] This means the curve intersects itself everywhere, forming a loop.

Determine the points where ( r = 0 ) to establish the boundaries of integration: [ 3\sin(2\theta  \frac{2\pi}{4})  \theta = 0 ] [ \sin(2\theta  \frac{2\pi}{4}) = \frac{\theta}{3} ] [ 2\theta  \frac{2\pi}{4} = \arcsin(\frac{\theta}{3}) ] [ \theta = \frac{\arcsin(\frac{\theta}{3})}{2} + \frac{\pi}{8} ]

Set up the integral for the area enclosed by the curve: [ A = \frac{1}{2}\int_{\frac{\pi}{8}}^{\frac{3\pi}{4}}(r)^2 , d\theta ]

Substitute the equation for ( r ) into the integral: [ A = \frac{1}{2}\int_{\frac{\pi}{8}}^{\frac{3\pi}{4}}(3\sin(2\theta  \frac{2\pi}{4})  \theta)^2 , d\theta ]

Evaluate the integral to find the area enclosed by the curve within the given range of ( \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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