Line Integrals for Areas

In this topic:

    (1) If is a region bounded by a piecewise smooth simple closed curve oriented counterclockwise, then the area of is given by


    
    (2) Use a line integral to find the area enclosed by the region defined by the circle
    
    (3) Use a line integral to find the area enclosed by the region defined by the triangle with vertices
    
    (4) Use a line integral to find the area enclosed by the region defined by the curve for
    

Proposition (Line Integral for Area) If is a region bounded by a piecewise smooth simple closed curve oriented counterclockwise, then the area of is given by

Example (Line Integral for Area) Use a line integral to find the area enclosed by the region defined by the circle

    Solution. We can parametrize the circle by and for Then the area is found by,
    




Example (Line Integral for Area) Use a line integral to find the area enclosed by the region defined by the triangle with vertices

    Solution. We can parametrize the line segments by
    






Then the area is found by,
    









Example (Line Integral for Area) Use a line integral to find the area enclosed by the region defined by the curve for

    Solution. We have,
    






















by using the reduction formula,

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Cite this as:
Line Integrals For Areas
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/line-integrals-for-areas.html