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Area Between Curves

Proposition (Area Between Curves) If area between curves _gr_1.gif] and area between curves _gr_2.gif] are continuous with area between curves _gr_3.gif] on the closed interval area between curves _gr_4.gif] then the area between the two curves area between curves _gr_5.gif] and area between curves _gr_6.gif] is given by

area between curves _gr_7.gif]

Example (Area Between Curves) Sketch the region bounded by area between curves _gr_8.gif] area between curves _gr_9.gif] area between curves _gr_10.gif] and area between curves _gr_11.gif] then find the area between the two curves.

    Solution. A sketch of the region is

area between curves _gr_12.gif]

The points of intersection are area between curves _gr_13.gif] and area between curves _gr_14.gif] Notice that area between curves _gr_15.gif] on area between curves _gr_16.gif] and therefore the area between the curves is (using symmetry)  

area between curves _gr_17.gif]

area between curves _gr_18.gif]

area between curves _gr_19.gif]

area between curves _gr_20.gif]

area between curves _gr_21.gif]
area between curves _gr_22.gif]

Example (Area Between Curves) Sketch the region bounded by area between curves _gr_23.gif] area between curves _gr_24.gif] area between curves _gr_25.gif] and area between curves _gr_26.gif] then find the area between the two curves.

    Solution. A sketch of the region is

area between curves _gr_27.gif]
The point of intersection is area between curves _gr_28.gif] Notice that area between curves _gr_29.gif] on area between curves _gr_30.gif] and therefore the area between the curves is

area between curves _gr_31.gif]

area between curves _gr_32.gif]

area between curves _gr_33.gif]

area between curves _gr_34.gif]

area between curves _gr_35.gif]

area between curves _gr_36.gif]

Example (Area Between Curves) Sketch the region bounded by area between curves _gr_37.gif] and area between curves _gr_38.gif] then find the area between the two curves.

    Solution. A sketch of the region is

area between curves _gr_39.gif]

The point of intersection is found by solving area between curves _gr_40.gif] So,

area between curves _gr_41.gif]

area between curves _gr_42.gif]

area between curves _gr_43.gif]

area between curves _gr_44.gif]

Thus the intersection points are area between curves _gr_45.gif] and area between curves _gr_46.gif] Notice that area between curves _gr_47.gif] on area between curves _gr_48.gif] and therefore the area between the curves is

area between curves _gr_49.gif]

area between curves _gr_50.gif]

area between curves _gr_51.gif]

area between curves _gr_52.gif]

area between curves _gr_53.gif]

area between curves _gr_54.gif]

area between curves _gr_55.gif]

Example (Area Between Curves) Sketch the region bounded by area between curves _gr_56.gif] and area between curves _gr_57.gif] then find the area between the two curves.

    Solution.  A sketch of the region is

area between curves _gr_58.gif]

The point of intersection is found by solving area between curves _gr_59.gif] So,

area between curves _gr_60.gif]

area between curves _gr_61.gif]

area between curves _gr_62.gif]

Thus the intersection points are area between curves _gr_63.gif] and area between curves _gr_64.gif] Notice that area between curves _gr_65.gif] on area between curves _gr_66.gif] and therefore the area between the curves is

area between curves _gr_67.gif]

area between curves _gr_68.gif]

area between curves _gr_69.gif]

area between curves _gr_70.gif]

area between curves _gr_71.gif]

area between curves _gr_72.gif]

Cite this as:
Area Between Curves
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/area-between-curves.html
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