CK12 Calculus - Single Variable

Now let’s computethe definiteintegralusingour definitionand also someof our summationformulas.
Example2:

Use the definitionof the definiteintegralto evaluate
Solution:
Applyingour definition,we needto find

=

We will use right endpointsto computethe integral.We first needto divide into sub-intervalsof

length Sincewe are usingright endpoints,

So

Recallthat. By substitution,we have

as.
Hence

Beforewe look to try someproblems,let’s makea coupleof observations.First,we will soonnot needto
rely on the summationformulaand RiemannSumsfor actualcomputationof definiteintegrals.We will develop
severalcomputationalstrategiesin orderto solvea varietyof problemsthat comeup. Second,the idea of
definiteintegralsas approximatingthe area undera curvecan be a bit confusingsincewe may sometimes
get resultsthat do not makesensewheninterpretedas areas.For example,if we wereto computethe

definiteintegral then due to the symmetryof aboutthe origin,we wouldfind that

This is becausefor everysamplepoint we also have is also a samplepointwith

Hence,it is moreaccurateto say that givesus the net area between

and If we wantedthe totalareaboundedby the graphand the -axis,then we would