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Equations of Curves are in parametric Form

It is known that the area between the curve and the lines , and is given by

.

If the equation of the curve is in parametric form

,

where t is a parameter, and if

is a continuous function on , and does not change sign is in , then the area of the region bounded by the curve , the x-axis and the lines , is

.

Example Find the area of the ellipse , where , .

Computing Arc Lengths

Equations of curves are in Rectangular Form

Theorem If a curve has a continuous derivative on , then the length of the curve from to is given by

.

If the equation of the curve is in the form , then the length of the arc between and is given by

.

Equations of Curves are in Parametric Form

Theorem When a function is expressed in parametric form and , the arc length of the curve from to is given by

.

Lecture 38 computing volumes and area of surfaces of revolution