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

This cannot be solved manually .

By signing up, you agree to our Terms of Service and Privacy Policy

To find the area enclosed by the curve ( r = \theta \cos(\theta) - 2 \sin\left(\frac{\theta}{2} - \pi\right) ) for ( \theta ) in ( [\frac{\pi}{4}, \pi] ), you need to integrate ( \frac{1}{2} r^2 ) with respect to ( \theta ) over the given interval. The integral will give you the area of one loop of the curve. Then, you can subtract the area under the curve for ( \theta ) from ( \frac{\pi}{4} ) to ( \frac{\pi}{2} ) from the area under the curve for ( \theta ) from ( \frac{\pi}{2} ) to ( \pi ). This subtraction accounts for the loops in opposite directions.

The expression for the area ( A ) enclosed by the curve is:

[ A = \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \frac{1}{2} \left(\theta \cos(\theta) - 2 \sin\left(\frac{\theta}{2} - \pi\right)\right)^2 , d\theta - \int_{\frac{\pi}{2}}^{\pi} \frac{1}{2} \left(\theta \cos(\theta) - 2 \sin\left(\frac{\theta}{2} - \pi\right)\right)^2 , d\theta ]

You can then calculate each integral separately using numerical methods or appropriate techniques like integration by parts or substitution.

By signing up, you agree to our Terms of Service and Privacy Policy

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.

- What is the Cartesian form of #( -7 , (-35pi)/12 ) #, where the original coordinates are in polar?
- What is the arclength of the polar curve #f(theta) = -5costheta+sin6theta # over #theta in [0,pi/3] #?
- How do you find the area the region of the common interior of #r=a(1+costheta), r=asintheta#?
- What is the Cartesian form of #( -2, (-7pi)/8 ) #?
- What is the polar form of #( -23,-3 )#?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7