Evaluate where C is the portion of the circle centered at the origin with radius 2 in the 1st quadrant with counter clockwise rotation and 1 F(z, y) = (2ry-4-,sin(x/2)sin(y/2)) 7+(x²+,cos(x/2)cos(y/2))7.

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A: Solution:-

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A: on solving this we get

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A: We have to find the tangent line

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A: Explanation and solutions is given below....

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A: In this we have to verify green theorem.

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A: Find the intersection points:

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A: The unit circle can be written in parametric form as x=cosθ and y=sinθ We have to find dydx and the…

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A: We need to find the length of the circle which is parametrically defined as below.

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A: The curve will appear like this

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A: Here I am attaching image so that you understand each and every step.

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Q: 13. Find the area of the surface generated when the curve = , on [2,4] is rotated about the y-axis.

A: The solution are next step is

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A: This is a application of integral problem

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Q: 12. Find the area of the surface generated when the curve y =+,on [1,2] is rotated about the x-axis.

A: solution is given below...

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Q: Find the arc length of the curve y = In(cos x),0 < x <. 3

A: Arc length of a curve y=f(x) for a≤x≤b is given by L=∫ab1+f'(x)2 dx

Q: Find the arclength of the curve x 5 cos(8t), y : 5 sin(8t) with 0 <t <

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A: Solution