Riemann Integration lecture 4||SyBsc Calculus || 7 September

Riemann Integration
Approximation of area, upper and lower Riemann sums and their properties, upper and lower integrals, definition of Riemann integral on a closed and bounded interval, Riemann criterion for integrability.
Indefinite and improper integrals
Fundamental theorem of calculus, Mean value theorem, Integration by parts, Leibnitz rule, Improper integrals-type 1 and type 2, Absolute convergence of improper integrals, Comparison tests, Abel's and Dirichlet's tests.
Applications
and functions and their properties, relationship between and functions (without proof).
Applications of definite Integrals: Area between curves, finding volumes by slicing, volumes of solids of revolution-Disks and Washers, Cylindrical Shells, Lengths of plane curves, Areas of surfaces of revolution.