Unit 7A

Integrals Part 1

Lesson 3: Areas Part 1

The method of exhaustion is closely related to the techniques used for finding areas in integral calculus.

Watch the video The Rectangular Method for Area to see how the method of exhaustion can be applied to find the area under a curve between two limiting values, «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«mo»=«/mo»«mi»a«/mi»«/math» and «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«mo»=«/mo»«mi»b«/mi»«/math».

In the video, the rectangular approximation method for finding the area under a curve was presented.  


To develop a formula to find the area under «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«mo»=«/mo»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/math» from «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»a«/mi»«/math» to «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»b«/mi»«/math» where «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«mo»=«/mo»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/math» is continuous and positive, subdivide the region into «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»n«/mi»«/math» strips of equal width.


From the diagram, the right-hand endpoints of the intervals are
     

The right-hand endpoint of the «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathsize=¨16px¨»i«/mi»«/math» th interval is


The height of the «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»i«/mi»«/math» th rectangle is «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mfenced»«msub»«mi»x«/mi»«mn»1«/mn»«/msub»«/mfenced»«/math», so its area is


And, if the widths get so small they approach zero (in other words there are infinitely many rectangles used), the exact area under the curve can be found. Therefore, to find the area under a curve, take the limit of the sum of the areas of the rectangles.


In addition to the limit expression used to find area, introduced above, there are some additional  summation notation formulas that are needed. They are as follows.

sum of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»1«/mn»«/math»: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«munderover»«mo»§#8721;«/mo»«mrow»«mi»i«/mi»«mo»=«/mo»«mi»i«/mi»«/mrow»«mi»n«/mi»«/munderover»«mn»1«/mn»«mo»=«/mo»«mi»n«/mi»«/math»
sum of a constant: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«munderover»«mo»§#8721;«/mo»«mrow»«mi»i«/mi»«mo»=«/mo»«mi»i«/mi»«/mrow»«mi»n«/mi»«/munderover»«mi»c«/mi»«mo»=«/mo»«mi»n«/mi»«mi»c«/mi»«/math» 
sum of the natural numbers: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«munderover»«mo»§#8721;«/mo»«mrow»«mi»i«/mi»«mo»=«/mo»«mi»i«/mi»«/mrow»«mi»n«/mi»«/munderover»«mi»i«/mi»«mo»=«/mo»«mfrac»«mrow»«mi»n«/mi»«mfenced»«mrow»«mi»n«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«/mrow»«mn»2«/mn»«/mfrac»«/math»
sum of the squares of the natural numbers: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«munderover»«mo»§#8721;«/mo»«mrow»«mi»i«/mi»«mo»=«/mo»«mi»i«/mi»«/mrow»«mi»n«/mi»«/munderover»«msup»«mi»i«/mi»«mn»2«/mn»«/msup»«mo»=«/mo»«mfrac»«mrow»«mi»n«/mi»«mfenced»«mrow»«mi»n«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mfenced»«mrow»«mn»2«/mn»«mi»n«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«/mrow»«mn»6«/mn»«/mfrac»«/math»
sum of the cubes of the natural numbers: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«munderover»«mo»§#8721;«/mo»«mrow»«mi»i«/mi»«mo»=«/mo»«mi»i«/mi»«/mrow»«mi»n«/mi»«/munderover»«msup»«mi»i«/mi»«mn»3«/mn»«/msup»«mo»=«/mo»«mfrac»«mrow»«msup»«mi»n«/mi»«mn»2«/mn»«/msup»«msup»«mfenced»«mrow»«mi»n«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mrow»«mn»4«/mn»«/mfrac»«/math»

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