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Definitions

Forman Definition: The area between the graph of y = f(x) and the x-axis is given by the definite integral below. This formula gives a positive result for a graph above the x-axis, and a negative result for a graph below the x-axis.
http://www.mathwords.com/a/area_under_a_curve.htm

Area Under a Curve (with respect to x): the area bounded by two curves and left to right bounds on the x-axis; the actual area is found by subtracting the bottom curve from the top curve and integrating that quantity using either the givin x-bounds or by using the x coordinates of the points of intersect for those two curves as your bounds


Area Under a Curve (with respect to y): the area bounded by two curves and bottom to top bounds on the y-axis; the actual area is found by subtracting the left most curve from the right most curve and integrating that quantity using either the given y-bounds or by using the y cooridinates of the points of intersect for those two curves as your bounds


Formulas and Tips

Finding area in respect to the x-axis.
form_1.gif
graph8.jpg


Finding area in respect to the y-axis

form_2.gif
graph9.jpg

To find the area of these functions using the y-axis you must use two different integrals hence the two colors shown.


It is usually easier to find area in respect to the x-axis because the functions are usually given as f(x) and you also have to find the function in respect to the x-axis if you need to graph it anyway


If you are going to solve in respect to the y-axis don't forget to solve for x instead of y
.


Steps

  1. Graph the functions
  2. Determine whether you are going to solve in respect to x-axis or y-axis
  3. Find the points of intersection
  4. Determine top and bottom curve (with respect to x-axis)/determine right-most and left-most curve (with respect to y-axis)
  5. Integrate top/right minus bottom/left

Videos



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Examples

Problem 1

Problem2.jpg
Solutionpic.jpg


http://apcentral.collegeboard.com/apc/members/exam/exam_questions/157006.html

Problem 2

Find the area of the region enclosed between the curves equation10.jpg and equation73.jpg

We first graph the two equations and examine the area enclosed between the curves.




area under curve, example 2
area under curve, example 2



The region whose area is in question is limited above by the curve equation73.jpg and below by the curve equation10.jpg. The left endpoint and the right endpoint of the region are the point of intersection of the curves and can be found by equating equation10.jpg and equation73.jpg and solving for x.
equation10.jpg


Which gives. equation62.jpg

or

(x + 1)(x - 2) = 0

which gives solutions

x = -1 and x = 2

Between the points of intersection, equation73.jpg is greater than or equal to equation10.jpg. Let equation37.jpg and equation83.jpg and apply formula 1 above to find the area A of the region between the two curves. The limits of integration are the x coordinates of the points of intersection found above: -1 and 2

A = Integral.jpg

=
Integral78.jpg

= Integral33.jpg



Integral7.jpg


The area of the region enclosed between the curves equation10.jpg and equation73.jpg is equal to 9.

http://www.analyzemath.com/calculus/Integrals/area_under_curve.html

Problem 3



problem_3.jpg
Graph.jpg
solution_3.jpg
Area.jpg


http://apcentral.collegeboard.com/apc/public/repository/ap09_frq_calculus_ab.pdf


Links

http://archives.math.utk.edu/visual.calculus/5/area2curves.2/index.html
http://cow.temple.edu/~cow/cgi-bin/manager
click: book two
applications of integration
area
area between two curves one/two
http://www.interactmath.com/asp/wizardframe.htm
Click: Book: Adams: Calculus, 6e ENHANCED
Chapter: 5. Integration
Section: 5.7 Areas of Plane Regions