Integration - the Australian Mathematical Sciences Institute

{20} • IntegrationExercise 8Find∫(3x 2 + 3 x) d x.Exercise 9Prove the theorem above (linearity of integration) using similar rules for differentiation.Calculating areas with antiderivativesA general method to find the area under a graph y = f (x) between x = a and x = b isgiven by the following important theorem.Theorem (Fundamental theorem of calculus)Let f (x) be a continuous real-valued function on the interval [a,b]. Then∫ ba[ bf (x) d x = F (x)]= F (b) − F (a),awhere F (x) is any antiderivative of f (x).The notation [ F (x) ] bais just a shorthand to substitute x = a and x = b into F (x) andsubtract; it is synonymous with F (b) − F (a).Note that any antiderivative of f (x) will work in the above theorem. Indeed, if we havetwo different antiderivatives F (x) and G(x) of f (x), then they must differ by a constant,so G(x) = F (x)+c for some constant c. Then we have to get the same answer whether weuse F or G, sinceG(b) −G(a) = ( F (b) + c ) − ( F (a) + c )= F (b) − F (a).We give a proof of the theorem, assuming our previous statements:• the derivative of the area function is f (x)• any two antiderivatives of f (x) differ by a constant.