What is the area enclosed by #r=cos(4theta-(7pi)/4)+sin(theta+(pi)/8) # between #theta in [pi/3,(5pi)/3]#?

Answer 1

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Answer 2

To find the area enclosed by the polar curve ( r = \cos(4\theta - \frac{7\pi}{4}) + \sin(\theta + \frac{\pi}{8}) ) between ( \theta ) in ( \left[\frac{\pi}{3}, \frac{5\pi}{3}\right] ), you can use the formula for the area of a polar region:

[ A = \frac{1}{2} \int_{\alpha}^{\beta} r^2 d\theta ]

where ( r ) is the polar function, and ( \alpha ) and ( \beta ) are the limits of integration for ( \theta ). Substituting the given function and limits, you can calculate the integral to find the area.

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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.