L4 Areas II - Part 1
Completion requirements
Unit 7A
Integrals Part 1
Lesson 4: Areas Part 2
Suppose you have the definite integral «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msubsup»«mo»§#8747;«/mo»«mrow»«mo»§#8722;«/mo»«mo»§#960;«/mo»«/mrow»«mo»§#960;«/mo»«/msubsup»«mo»§#8722;«/mo»«mi»sin«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨» «/mi»«mi»d«/mi»«mi»x«/mi»«/math». This can be evaluated as
Given the evaluation of the definite integral above, the area under the curve of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mo»§#8722;«/mo»«mi»sin«/mi»«mi»x«/mi»«/math» between «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8722;«/mo»«mo»§#960;«/mo»«/math» and «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨normal¨»§#960;«/mi»«/math» should be zero. However, as can be seen in the graph of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mo»§#8722;«/mo»«mi»sin«/mi»«mi»x«/mi»«/math» below, the area is clearly not zero. Then, why is the value of the integral zero? Try to reconcile this value with the graph shown.
How could the shaded area be accurately determined?
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«msubsup»«mfenced open=¨¨ close=¨|¨»«mrow»«mi»cos«/mi»«mi»x«/mi»«/mrow»«/mfenced»«mrow»«mo»§#8722;«/mo»«mo»§#960;«/mo»«/mrow»«mo»§#960;«/mo»«/msubsup»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»cos«/mi»«mfenced»«mo»§#960;«/mo»«/mfenced»«mo»§#8722;«/mo»«mi»cos«/mi»«mfenced»«mrow»«mo»§#8722;«/mo»«mo»§#960;«/mo»«/mrow»«/mfenced»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#8722;«/mo»«mn»1«/mn»«mo»§#8722;«/mo»«mfenced»«mrow»«mo»§#8722;«/mo»«mn»1«/mn»«/mrow»«/mfenced»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»0«/mn»«/mtd»«/mtr»«/mtable»«/math»
Given the evaluation of the definite integral above, the area under the curve of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mo»§#8722;«/mo»«mi»sin«/mi»«mi»x«/mi»«/math» between «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8722;«/mo»«mo»§#960;«/mo»«/math» and «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨normal¨»§#960;«/mi»«/math» should be zero. However, as can be seen in the graph of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mo»§#8722;«/mo»«mi»sin«/mi»«mi»x«/mi»«/math» below, the area is clearly not zero. Then, why is the value of the integral zero? Try to reconcile this value with the graph shown.
How could the shaded area be accurately determined?