Learning Objectives Upon completion of this Chapter, students will be able to • define a definite integral as the limit of a sum • demonstrate a definite integral geometrically • use the fundamental theorem of integral calculus • evaluate definite integrals by evaluating anti-derivatives • establish some properties of definite integrals • identify improper integrals and use the gamma integral • use reduction formulae • apply definite integral to evaluate area of a plane region • apply definite integral to evaluate the volume of a solid of revolution We briefly recall what we have already studied about anti-derivative of a given function f x ( ) . If a function F x ( ) can be found such that d dx F x , then the function F x ( ) is called an anti-derivative of f x ( ) . Fig. .
Archimedes of Syracuse (288BC(BCE)-212BC(BCE)) was a Greek mathematician, physicist, engineer, inventor B ( , ) C , ) − A − − O of of Applications of Integration It is not unique, because, for any arbitrary constant C , we get d dx F x d dx F x [ ( ) ] [ ( )] That is, if F x ( ) is an anti-derivative of f x ( ) , then the function F x ( ) + is also an anti-derivative of the same function f x ( ) . Note that all anti-derivatives of f x ( ) differ by a constant only. The anti-derivative of f x ( ) is usually called the indefinite integral of f x ( ) with respect to x and is denoted by