Parametric and polar integration
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Parametric and Polar Integration. Area Enclosed Parametrically.

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Parametric and Polar Integration

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Parametric and polar integration

Parametric and Polar Integration

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)


Parametric and polar integration

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 and polar integration

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.

Parametric and polar integration

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.

Parametric and polar integration

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:

Parametric and polar integration

Polar Area

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

Parametric and polar integration

Polar Area

Example: Find the area enclosed by:


Specifically a Cardiod)

Parametric and polar integration

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.

Parametric and polar integration

Polar Area

Example: Find the area inside and outside

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

Parametric and polar integration

When finding area, negative values of r cancel out:

Area of one leaf times 4:

Area of four leaves:

Parametric and polar integration

To find the length of a curve:


Again, for polar graphs:

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


Parametric and polar integration

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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