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Page 1 DEFINITE INTEGRAL Here we shall define integration as a process of summation or define integral as the limit of a sum. Then we discuss some properties of definite integral. The concept of definite integral is then used to find the area enclosed by certain curves. Page 2 DEFINITE INTEGRAL Here we shall define integration as a process of summation or define integral as the limit of a sum. Then we discuss some properties of definite integral. The concept of definite integral is then used to find the area enclosed by certain curves. Fundamental theorem of calculus Let ?? ( ?? ) be a continuous real valued function defined on [ ?? , ?? ] such that ? ?? ( ?? ) ???? = ?? ( ?? ) + ?? . Then ? ?? ?? ? ?? ( ?? ) ???? = ?? ( ?? ) - ?? ( ?? ), called definite integral of ?? ( ?? ) in [ ?? , ?? ]. Page 3 DEFINITE INTEGRAL Here we shall define integration as a process of summation or define integral as the limit of a sum. Then we discuss some properties of definite integral. The concept of definite integral is then used to find the area enclosed by certain curves. Fundamental theorem of calculus Let ?? ( ?? ) be a continuous real valued function defined on [ ?? , ?? ] such that ? ?? ( ?? ) ???? = ?? ( ?? ) + ?? . Then ? ?? ?? ? ?? ( ?? ) ???? = ?? ( ?? ) - ?? ( ?? ), called definite integral of ?? ( ?? ) in [ ?? , ?? ]. Remark: 1 If ?? ( ?? ) is discontinuous at ?? = ?? and continuous at ?? = ?? then ? ?? ?? ? ?? ( ?? )???? = ?? ( ?? ) - l im ?? ? ?? + ? ?? ( ?? ) 2 If ?? ( ?? ) is discontinuous at ?? = ?? and continuous at ?? = ?? then ? ?? ?? ? ?? ( ?? )???? = l im ?? ? ?? - ? ?? ( ?? ) - ?? ( ?? ) Page 4 DEFINITE INTEGRAL Here we shall define integration as a process of summation or define integral as the limit of a sum. Then we discuss some properties of definite integral. The concept of definite integral is then used to find the area enclosed by certain curves. Fundamental theorem of calculus Let ?? ( ?? ) be a continuous real valued function defined on [ ?? , ?? ] such that ? ?? ( ?? ) ???? = ?? ( ?? ) + ?? . Then ? ?? ?? ? ?? ( ?? ) ???? = ?? ( ?? ) - ?? ( ?? ), called definite integral of ?? ( ?? ) in [ ?? , ?? ]. Remark: 1 If ?? ( ?? ) is discontinuous at ?? = ?? and continuous at ?? = ?? then ? ?? ?? ? ?? ( ?? )???? = ?? ( ?? ) - l im ?? ? ?? + ? ?? ( ?? ) 2 If ?? ( ?? ) is discontinuous at ?? = ?? and continuous at ?? = ?? then ? ?? ?? ? ?? ( ?? )???? = l im ?? ? ?? - ? ?? ( ?? ) - ?? ( ?? ) Remark: 3. If ?? ( ?? ) is discontinuous at ?? = ?? and ?? = ?? then ? ?? ?? ? ?? ( ?? ) ???? = l im ?? ? 0 ? ? ?? + ?? ?? - ?? ? ?? ( ?? )???? or l im ?? ? ?? - ? ?? ( ?? ) - l im ?? ? ?? + ? ?? ( ?? ) 4. If ?? ( ?? ) is discontinuous at ?? = ?? ( ?? < ?? < ?? ) then ? ?? ?? ? ?? ( ?? )???? = l im ?? ? 0 ? ? ?? ?? - ?? ? ?? ( ?? )???? + l im ?? ? 0 ? ? ?? + ?? ?? ? ?? ( ?? ) ???? Page 5 DEFINITE INTEGRAL Here we shall define integration as a process of summation or define integral as the limit of a sum. Then we discuss some properties of definite integral. The concept of definite integral is then used to find the area enclosed by certain curves. Fundamental theorem of calculus Let ?? ( ?? ) be a continuous real valued function defined on [ ?? , ?? ] such that ? ?? ( ?? ) ???? = ?? ( ?? ) + ?? . Then ? ?? ?? ? ?? ( ?? ) ???? = ?? ( ?? ) - ?? ( ?? ), called definite integral of ?? ( ?? ) in [ ?? , ?? ]. Remark: 1 If ?? ( ?? ) is discontinuous at ?? = ?? and continuous at ?? = ?? then ? ?? ?? ? ?? ( ?? )???? = ?? ( ?? ) - l im ?? ? ?? + ? ?? ( ?? ) 2 If ?? ( ?? ) is discontinuous at ?? = ?? and continuous at ?? = ?? then ? ?? ?? ? ?? ( ?? )???? = l im ?? ? ?? - ? ?? ( ?? ) - ?? ( ?? ) Remark: 3. If ?? ( ?? ) is discontinuous at ?? = ?? and ?? = ?? then ? ?? ?? ? ?? ( ?? ) ???? = l im ?? ? 0 ? ? ?? + ?? ?? - ?? ? ?? ( ?? )???? or l im ?? ? ?? - ? ?? ( ?? ) - l im ?? ? ?? + ? ?? ( ?? ) 4. If ?? ( ?? ) is discontinuous at ?? = ?? ( ?? < ?? < ?? ) then ? ?? ?? ? ?? ( ?? )???? = l im ?? ? 0 ? ? ?? ?? - ?? ? ?? ( ?? )???? + l im ?? ? 0 ? ? ?? + ?? ?? ? ?? ( ?? ) ???? Note that even if ?? ( ?? ) is not defined at ?? = ?? or ?? = ?? or at both, ? ?? ?? ? ?? ( ?? )???? can be evaluated. ?? and ?? are called lower and upper limits of integration respectively. If we make change in variable (i.e. substitution) then limit of integration should be changed accordingly.Read More
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