What is the area under the polar curve #f(theta) = 3theta^2+thetasin(4theta-(5pi)/12 ) +cos(2theta-(pi)/3)# over #[pi/8,pi/6]#?

Answer 1

0.455

Area under the polar curve for any function #f(theta)# is given by #int_a^b 1/2 r^2 d(theta)#
The function #f(theta)# represents the r or radius as the curve moves from #pi/8# to #pi/6#.
So all we have to do is plugin whole #f(theta)# for r.
That would give us, #int_(pi/8)^(pi/6) (3(theta)^2 +theta*sin(4theta-((5pi)/12))+cos(2theta-pi/3))^2d(theta)#

Doing this integral by hand would be little tedious so you can just plug integral in your calculator and get the answer.

The answer would be 0.455

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Answer 2

To find the area under the polar curve ( f(\theta) = 3\theta^2 + \theta\sin(4\theta - \frac{5\pi}{12}) + \cos(2\theta - \frac{\pi}{3}) ) over the interval ( \left[\frac{\pi}{8}, \frac{\pi}{6}\right] ), follow these steps:

  1. Compute the definite integral of ( f(\theta) ) over the given interval.
  2. Substitute the upper limit of integration into ( f(\theta) ) and subtract the result of substituting the lower limit of integration.
  3. Evaluate the resulting expression.

This will give you the area under the polar curve over the specified interval.

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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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