Unit 7A

Integrals Part 1

Lesson 3: Areas Part 1

Fundamental Theorem of Calculus

If «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 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», then
 
«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»«mo»=«/mo»«mi»F«/mi»«mfenced»«mi»b«/mi»«/mfenced»«mo»§#8722;«/mo»«mi»F«/mi»«mfenced»«mi»a«/mi»«/mfenced»«/math»,

where «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»F«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/math» is an antiderivative of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/math».

Additional notation
As with most notations in math, there is more than one way of showing the evaluation of the definite integral. 

«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»«mi»F«/mi»«mfenced»«mi»b«/mi»«/mfenced»«mo»§#8722;«/mo»«mi»F«/mi»«mfenced»«mi»a«/mi»«/mfenced»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»F«/mi»«msubsup»«mfenced open=¨¨ close=¨]¨»«mfenced»«mi»x«/mi»«/mfenced»«/mfenced»«mi»a«/mi»«mi»b«/mi»«/msubsup»«mo»§#160;«/mo»«mi mathvariant=¨normal¨» «/mi»«mi»or«/mi»«mi mathvariant=¨normal¨» «/mi»«mo»§#160;«/mo»«mi»F«/mi»«msubsup»«mfenced open=¨¨ close=¨|¨»«mfenced»«mi»x«/mi»«/mfenced»«/mfenced»«mi»a«/mi»«mi»b«/mi»«/msubsup»«/mtd»«/mtr»«/mtable»«/math»

Find the exact value of the area bounded by the curve «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mn»4«/mn»«mo»§#8722;«/mo»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/math» and the «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«/math»-axis. Note: This is the same question as Example 1. However, the Fundamental Theorem of Calculus will be used to find the solution this time.




The desired area extends from «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«mo»=«/mo»«mo»§#8722;«/mo»«mn»2«/mn»«/math» to «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«mo»=«/mo»«mn»2«/mn»«/math». Using the Fundamental Theorem of Calculus to determine the area, it is not necessary to simplify the problem by initially only finding half the area as was done in Example 1.

First, determine the indefinite integral of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/math» (the general antiderivative).

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mo»§#8747;«/mo»«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»«mi»F«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»+«/mo»«mi»C«/mi»«/mtd»«/mtr»«mtr»«mtd»«mo»§#8747;«/mo»«mfenced»«mrow»«mn»4«/mn»«mo»§#8722;«/mo»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfenced»«mi mathvariant=¨normal¨» «/mi»«mi»d«/mi»«mi»x«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»4«/mn»«mi»x«/mi»«mo»§#8722;«/mo»«mfrac»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mn»3«/mn»«/mfrac»«mo»+«/mo»«mi»C«/mi»«/mtd»«/mtr»«/mtable»«/math»

Now, evaluate the definite integral to determine the area.

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«msubsup»«mo»§#8747;«/mo»«mrow»«mo»§#8722;«/mo»«mn»2«/mn»«/mrow»«mn»2«/mn»«/msubsup»«mfenced»«mrow»«mn»4«/mn»«mo»§#8722;«/mo»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfenced»«mi»d«/mi»«mi»x«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»F«/mi»«mi mathvariant=¨normal¨»(«/mi»«mn»2«/mn»«mi mathvariant=¨normal¨»)«/mi»«mo»§#8722;«/mo»«mi»F«/mi»«mfenced»«mrow»«mo»§#8722;«/mo»«mn»2«/mn»«/mrow»«/mfenced»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfenced»«mrow»«mn»4«/mn»«mfenced»«mn»2«/mn»«/mfenced»«mo»§#8722;«/mo»«mfrac»«msup»«mfenced»«mn»2«/mn»«/mfenced»«mn»3«/mn»«/msup»«mn»3«/mn»«/mfrac»«mo»+«/mo»«mi»C«/mi»«/mrow»«/mfenced»«mo»§#8722;«/mo»«mfenced»«mrow»«mn»4«/mn»«mfenced»«mrow»«mo»§#8722;«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mo»§#8722;«/mo»«mfrac»«msup»«mfenced»«mrow»«mo»§#8722;«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mn»3«/mn»«/msup»«mn»3«/mn»«/mfrac»«mo»+«/mo»«mi»C«/mi»«/mrow»«/mfenced»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfenced»«mrow»«mn»8«/mn»«mo»§#8722;«/mo»«mfrac»«mn»8«/mn»«mn»3«/mn»«/mfrac»«mo»+«/mo»«mi»C«/mi»«/mrow»«/mfenced»«mo»§#8722;«/mo»«mfenced»«mrow»«mo»§#8722;«/mo»«mn»8«/mn»«mo»+«/mo»«mfrac»«mn»8«/mn»«mn»3«/mn»«/mfrac»«mo»+«/mo»«mi»C«/mi»«/mrow»«/mfenced»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»8«/mn»«mo»§#8722;«/mo»«mfrac»«mn»8«/mn»«mn»3«/mn»«/mfrac»«mo»+«/mo»«mi»C«/mi»«mo»+«/mo»«mn»8«/mn»«mo»§#8722;«/mo»«mfrac»«mn»8«/mn»«mn»3«/mn»«/mfrac»«mo»§#8722;«/mo»«mi»C«/mi»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»16«/mn»«mo»§#8722;«/mo»«mfrac»«mn»16«/mn»«mn»3«/mn»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mn»32«/mn»«mn»3«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«/math»

Notice the constant «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»C«/mi»«/math» is eliminated by subtraction. Since this will always occur, from now on  - when calculating definite integrals, do NOT include the constant of integration, C.

This is the same value obtained in Example 1, when the area was determined as the limit of the sum of the areas of inscribed rectangles, where the number of rectangles increased without bound.