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Parametric and Polar Integration. Area Enclosed Parametrically.

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area enclosed parametrically
Area Enclosed Parametrically
  • Suppose that the parametric equations x = x(t) and y = y(t) with c  t  d, describe a curve that is traced out clockwise exactly once, as t increases from c to d and where the curve does not intersect itself, except that the initial and terminal points are the same. Then, the enclosed area is given by
  • If the curve is traced out counterclockwise, then the enclosed area is given by
area parametrically
Area Parametrically
  • Example:

Find the area enclosed by the path of the Scrambler (a popular carnival ride) if its path is represented by

(Curve is traced out counterclockwise once for 0  t  2)



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)


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


Find the distance traveled by the point in one complete trip about the larger circle.


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

surface area
Surface Area
  • Find the surface area of the surface formed by revolving the curve and for
  • about the line x = 2.

Area Inside a Polar Graph:

The length of an arc (in a circle) is given by r.q when q is given in radians.

For a very small q, the curve could be approximated by a straight line and the area could be found using the triangle formula:


Polar Area

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


Polar Area

Example: Find the area enclosed by:


Specifically a Cardiod)


Polar Area


To find the area between curves, subtract:

Just like finding the areas between Cartesian curves, establish limits of integration where the curves cross.


Polar Area

Example: Find the area inside and outside

To establish bounds, we must find where the two curves intersect.


When finding area, negative values of r cancel out:

Area of one leaf times 4:

Area of four leaves:


To find the length of a curve:


Again, for polar graphs:

If we find derivatives and plug them into the formula, we (eventually) get:



There is also a surface area equation similar to the others we are already familiar with:

When rotated about the x-axis: