L3 Areas I - Part 4
Completion requirements
Unit 7A
Integrals Part 1
Lesson 3: Areas Part 1
The method of exhaustion with limits applied works to find the area beneath a curve with set endpoints (left and right). However, that method is not particularly efficient, especially as the functions involved become more complex. As such, it is time
to explore another, more efficient approach. This new method involves the definite integral and the Fundamental Theorem of Calculus.
This formula was derived from finding the sum of the areas of rectangles under a curve «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»x«/mi»«mo»=«/mo»«mi»a«/mi»«/math» to «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«mo»=«/mo»«mi»b«/mi»«/math».
When a definite integral is used to determine the sum of the areas of rectangles under a positive and continuous curve «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»x«/mi»«mo»=«/mo»«mi»a«/mi»«/math» to «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«mo»=«/mo»«mi»b«/mi»«/math», the following notation can be used.
The definite integral is used to represent the area bounded by a positive and continuous function «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«mo»=«/mo»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/math», the line «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«mo»=«/mo»«mi»a«/mi»«/math» on the left, the line «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«mo»=«/mo»«mi»b«/mi»«/math» on the right, and the «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«/math»-axis.
In the notation «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msubsup»«mo»§#8747;«/mo»«mi»a«/mi»«mi»b«/mi»«/msubsup»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mi»d«/mi»«mi»x«/mi»«/math», «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/math» is called the integrand and «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»a«/mi»«/math» and «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»b«/mi»«/math» are called the limits of integration, where «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»a«/mi»«/math» is the lower bound or lower limit and is the upper bound or upper limit. The notation «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msubsup»«mo»§#8747;«/mo»«mi»a«/mi»«mi»b«/mi»«/msubsup»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mi»d«/mi»«mi»x«/mi»«/math» should be regarded as a single symbol.
Watch the video The Fundamental Theorem of Calculus Part I for a discussion of the connection between derivatives, antiderivatives, and the definite integral .
Also, watch the video The Fundamental Theorem of Calculus Part II to discover the connection between the definite integral and one of its many antiderivatives. This part of the theorem simplifies the calculation of the definite integral.
The Definite Integral
In the previous section of this Lesson, area was written as a limit as follows.«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»Area«/mi»«mo»=«/mo»«munder»«mi»lim«/mi»«mrow»«mi»n«/mi»«mo»§#8594;«/mo»«mo»§#8734;«/mo»«/mrow»«/munder»«munderover»«mo»§#8721;«/mo»«mrow»«mi»i«/mi»«mo»=«/mo»«mn»1«/mn»«/mrow»«mi»n«/mi»«/munderover»«mi»f«/mi»«mfenced»«msub»«mi»x«/mi»«mi»i«/mi»«/msub»«/mfenced»«mo»§#916;«/mo»«mi»x«/mi»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mi»where«/mi»«mi mathvariant=¨normal¨» «/mi»«mo»§#160;«/mo»«mo»§#916;«/mo»«mi»x«/mi»«mo»=«/mo»«mfrac»«mrow»«mi»b«/mi»«mo»§#8722;«/mo»«mi»a«/mi»«/mrow»«mi»n«/mi»«/mfrac»«mo»§#160;«/mo»«mi mathvariant=¨normal¨» «/mi»«mi»and«/mi»«mi mathvariant=¨normal¨» «/mi»«mo»§#160;«/mo»«msub»«mi»x«/mi»«mi»i«/mi»«/msub»«mo»=«/mo»«mi»a«/mi»«mo»+«/mo»«mi»i«/mi»«mo»§#916;«/mo»«mi»x«/mi»«/math»
This formula was derived from finding the sum of the areas of rectangles under a curve «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»x«/mi»«mo»=«/mo»«mi»a«/mi»«/math» to «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«mo»=«/mo»«mi»b«/mi»«/math».
When a definite integral is used to determine the sum of the areas of rectangles under a positive and continuous curve «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»x«/mi»«mo»=«/mo»«mi»a«/mi»«/math» to «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«mo»=«/mo»«mi»b«/mi»«/math», the following notation can be used.
| Let «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«mo»=«/mo»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/math» be a function continuous on the interval «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfenced open=¨[¨ close=¨]¨»«mrow»«mi»a«/mi»«mo»,«/mo»«mo»§#160;«/mo»«mi»b«/mi»«/mrow»«/mfenced»«/math».
The definite integral of «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»
is as follows.
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«msubsup»«mo»§#8747;«/mo»«mi»a«/mi»«mi»b«/mi»«/msubsup»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mi»d«/mi»«mi»x«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«munder»«mi»lim«/mi»«mrow»«mi»n«/mi»«mo»§#8594;«/mo»«mo»§#8734;«/mo»«/mrow»«/munder»«munderover»«mo»§#8721;«/mo»«mrow»«mi»i«/mi»«mo»=«/mo»«mn»1«/mn»«/mrow»«mi»n«/mi»«/munderover»«mi»f«/mi»«mfenced»«msub»«mi»x«/mi»«mi»i«/mi»«/msub»«/mfenced»«mo»§#916;«/mo»«mi»x«/mi»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.125em¨/»«mspace width=¨0.125em¨/»«mspace width=¨0.125em¨/»«mspace width=¨0.125em¨/»«mspace width=¨0.125em¨/»«mspace width=¨0.125em¨/»«/mtd»«/mtr»«mtr»«mtd»«mi mathvariant=¨normal¨» «/mi»«mi»where«/mi»«mi mathvariant=¨normal¨» «/mi»«mo»§#160;«/mo»«mo»§#916;«/mo»«mi»x«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mi»b«/mi»«mo»§#8722;«/mo»«mi»a«/mi»«/mrow»«mi»n«/mi»«/mfrac»«mo»§#160;«/mo»«mi mathvariant=¨normal¨» «/mi»«mi»and«/mi»«mo»§#160;«/mo»«mi mathvariant=¨normal¨» «/mi»«msub»«mi»x«/mi»«mi»i«/mi»«/msub»«mo»=«/mo»«mi»a«/mi»«mo»+«/mo»«mi»i«/mi»«mo»§#916;«/mo»«mi»x«/mi»«/mtd»«/mtr»«/mtable»«/math»
|
The definite integral is used to represent the area bounded by a positive and continuous function «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«mo»=«/mo»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/math», the line «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«mo»=«/mo»«mi»a«/mi»«/math» on the left, the line «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«mo»=«/mo»«mi»b«/mi»«/math» on the right, and the «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«/math»-axis.
In the notation «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msubsup»«mo»§#8747;«/mo»«mi»a«/mi»«mi»b«/mi»«/msubsup»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mi»d«/mi»«mi»x«/mi»«/math», «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/math» is called the integrand and «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»a«/mi»«/math» and «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»b«/mi»«/math» are called the limits of integration, where «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»a«/mi»«/math» is the lower bound or lower limit and is the upper bound or upper limit. The notation «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msubsup»«mo»§#8747;«/mo»«mi»a«/mi»«mi»b«/mi»«/msubsup»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mi»d«/mi»«mi»x«/mi»«/math» should be regarded as a single symbol.
Watch the video The Fundamental Theorem of Calculus Part I for a discussion of the connection between derivatives, antiderivatives, and the definite integral .
Also, watch the video The Fundamental Theorem of Calculus Part II to discover the connection between the definite integral and one of its many antiderivatives. This part of the theorem simplifies the calculation of the definite integral.