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Arc Length Cartesian, Parametric, and Polar

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Arc Length x k-1 xkxk Green line = If we do this over and over from every x k—1 to any x k, we get

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Arc Length If we make x infinitely small, we have the Riemann Sum OR If y is a smooth function of x. If x is a smooth function of y.

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Arc Length Example: Without a calculator, find the arc length of the curve for 0 x 1.

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Arc Length Find the length of the curve between x = 8 and x = 8. Since y is NOT DIFFERENTIABLE between –8 and 8, we must use x in terms of y. fInt

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Surface Area The surface area of a solid of revolution depends on the radius (the distance between the graph and the axis of revolution) and the arc length.

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Surface Area Find the area of the surface formed by revolving the graph of on the interval [0, 1] about the x-axis. Use u-sub.

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Surface Area Find the area of the surface formed by revolving the graph of f(x) = x 2 on the interval about the y-axis. In this case, the r(x) is x since the axis of revolution is the y-axis.

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The equation for the length of a parametrized curve is similar to our previous “length of curve” equation: (Notice the use of the Pythagorean Theorem.) (proof on pg. 721)

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Parametric Arc Length A circle of radius 1 rolls around the circumference of a larger circle of radius 4. The epicycloid traced by a point on the circumference of the smaller circle is given by and Find the distance traveled by the point in one complete trip about the larger circle.

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Likewise, the equations for the surface area of a parametrized curve are similar to our previous “surface area” equations:

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The length of an arc (in a circle) is given by r. when is given in radians. Area Inside a Polar Graph: For a very small , the curve could be approximated by a straight line and the area could be found using the triangle formula:

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We can use this to find the area inside a polar graph.

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Example: Find the area enclosed by: This graph is called a lima ƈon.

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Notes: To find the area between curves, subtract: Just like finding the areas between Cartesian curves, establish limits of integration where the curves cross.

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When finding area, negative values of r cancel out: Area of one leaf times 4:Area of four leaves:

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To find the length of a curve: Remember: Again, for polar graphs: If we find derivatives and plug them into the formula, we (eventually) get: So:

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There is also a surface area equation similar to the others we are already familiar with: When rotated about the x-axis:

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