What is the area enclosed by #r=-3sin(2theta-(2pi)/4) -theta# between #theta in [pi/8,(3pi)/4]#?

Answer 1

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Answer 2

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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Answer 3

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}] ):

  1. 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.

  2. 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} ]

  3. 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 ]

  4. 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 ]

  5. Evaluate the integral to find the area enclosed by the curve within the given range of ( \theta ).

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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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