Answers
ApplicationsfromPhysics,Engineering,and Statistics
LearningObjectives
A studentwill be able to:
- Learnhow to applydefiniteintegralsto severalapplicationsfromphysics,engineering,and applied
mathematicssuchas work,fluidsstatics,and probability.
In this sectionwe will showhow the definiteintegralcan be usedin differentapplications.Someof the
conceptsmay soundnew to the reader, but we will explainwhatyou needto comprehendas we go along.
We will take threeapplications:The concepts,of workfrom physics,fluid staticsfrom engineering,and the
normalprobabilityfrom statistics.
Work
Work in physicsis definedas the productof the forceand displacement.Forceand displacementare vector
quantities,whichmeansthey havea directionand a magnitude.For example,we say the compressorexerts
a forceof Newtons upward.The magnitudehere is and the directionis upward.Lowering
a bookfrom an uppershelf to a lowerone by a distanceof metersawayfrom its initialpositionis another
exampleof the vectornatureof the displacement.Here,the magnitudeis m and the directionis downward,
usuallyindicatedby a minussign, i.e., a displacementof m. The productof thosetwo vectorquantities
(calledthe innerproduct, see Chapter10) givesthe workdoneby the force.Mathematically, we say
where is the forceand is the displacement.If the forceis measuredin Newtonsand distanceis in
meters,then workis measuredin the unitsof energywhichis in joules
Example1:
You pushan emptygrocerycart with a forceof for a distanceof meters.How muchworkis done
by you (the force)?