Parametric Curve and Arc length:

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print(); # input placeholder

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print(); # input placeholder

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print(); # input placeholder

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print(); # input placeholder

Surface Area:

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Find the surface area of a donut:

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print(); # input placeholder

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Top half of the surface area:JSFH

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print(); # input placeholder

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print(); # input placeholder

Bottom half of the surface area:

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print(); # input placeholder

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Total surface area is top+bottom=8Pi^2 or we can combine the top and bottom and use the following integration:

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print(); # input placeholder

Exercise. Can you find the surface of a cone with the height h and radius r by using the arc length concept?

Another example:

For part of the polar graph r = 2 + 3 cos(t), t is in [0; 2*Pi], part 1: set up the surface area by rotating the polar graph around x=-4

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part 2: set up the surface area by rotating the polar graph around y=4

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